Tabelle von Ableitungs- und Stammfunktionen

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Dieser Artikel ist eine Formelsammlung zum Thema Ableitungs- und Stammfunktionen. Es werden mathematische Symbole verwendet, die im Artikel Liste mathematischer Symbole erläutert werden.

Diese Tabelle von Ableitungs- und Stammfunktionen (Integraltafel) gibt eine Übersicht über Ableitungsfunktionen und Stammfunktionen, die in der Differential- und Integralrechnung benötigt werden.

Tabelle einfacher Ableitungs- und Stammfunktionen (Grundintegrale)

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Diese Tabelle ist zweispaltig aufgebaut. In der linken Spalte steht eine Funktion, in der rechten Spalte eine Stammfunktion dieser Funktion. Die Funktion in der linken Spalte ist somit die Ableitung der Funktion in der rechten Spalte.

Hinweise:

  • Wenn F {\displaystyle F} {\displaystyle F} eine Stammfunktion von f {\displaystyle f} {\displaystyle f} ist und C {\displaystyle C} {\displaystyle C} eine beliebige reelle Zahl (Konstante), dann ist auch F ( x ) + C {\displaystyle F(x)+C} {\displaystyle F(x)+C} eine Stammfunktion von f {\displaystyle f} {\displaystyle f}. Zum Beispiel ist auch F ( x ) = 1 2 x 2 + 5 {\displaystyle F(x)={\tfrac {1}{2}}x^{2}+5} {\displaystyle F(x)={\tfrac {1}{2}}x^{2}+5} eine Stammfunktion von f ( x ) = x {\displaystyle f(x)=x} {\displaystyle f(x)=x}. Ist der Definitionsbereich von f {\displaystyle f} {\displaystyle f} ein Intervall, so erhält man auf diese Art alle Stammfunktionen. Besteht der Definitionsbereich von f {\displaystyle f} {\displaystyle f} aus mehreren Intervallen, so kann die additive Konstante auf jedem der Intervalle getrennt gewählt werden. Die additive Konstante C {\displaystyle C} {\displaystyle C} wird aus Gründen der Übersichtlichkeit in der Tabelle nicht aufgeführt.
  • Weiterhin gilt: Falls F ( x ) {\displaystyle F(x)} {\displaystyle F(x)} eine Stammfunktion von f ( x ) {\displaystyle f(x)} {\displaystyle f(x)} ist, so ist aufgrund der Linearität des Integrals a ⋅ F ( x ) {\displaystyle a\cdot F(x)} {\displaystyle a\cdot F(x)} eine Stammfunktion von a ⋅ f ( x ) {\displaystyle a\cdot f(x)} {\displaystyle a\cdot f(x)}.
  • Ebenso gilt: Sind F ( x ) {\displaystyle F(x)} {\displaystyle F(x)} und G ( x ) {\displaystyle G(x)} {\displaystyle G(x)} Stammfunktionen von f ( x ) {\displaystyle f(x)} {\displaystyle f(x)} und g ( x ) {\displaystyle g(x)} {\displaystyle g(x)}, so ist F ( x ) + G ( x ) {\displaystyle F(x)+G(x)} {\displaystyle F(x)+G(x)} eine Stammfunktion von f ( x ) + g ( x ) {\displaystyle f(x)+g(x)} {\displaystyle f(x)+g(x)}.

Potenz- und Wurzelfunktionen

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Funktion f ( x ) {\displaystyle f(x)} {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} {\displaystyle F(x)}
0 {\displaystyle 0} {\displaystyle 0} 0 {\displaystyle 0} {\displaystyle 0}
k ( k ∈ R ) {\displaystyle k\quad (k\in \mathbb {R} )} {\displaystyle k\quad (k\in \mathbb {R} )} k x {\displaystyle kx} {\displaystyle kx}
x n {\displaystyle x^{n}} {\displaystyle x^{n}} { 1 n + 1 x n + 1 , wenn  n ≠ − 1 , ln ⁡ | x | , wenn  n = − 1. {\displaystyle {\begin{cases}{\frac {1}{n+1}}x^{n+1},&{\text{wenn }}n\neq -1,\\\ln |x|,&{\text{wenn }}n=-1.\end{cases}}} {\displaystyle {\begin{cases}{\frac {1}{n+1}}x^{n+1},&{\text{wenn }}n\neq -1,\\\ln |x|,&{\text{wenn }}n=-1.\end{cases}}}
n x n − 1 {\displaystyle nx^{n-1}} {\displaystyle nx^{n-1}} x n {\displaystyle x^{n}} {\displaystyle x^{n}}
x {\displaystyle x} {\displaystyle x} 1 2 x 2 {\displaystyle {\tfrac {1}{2}}x^{2}} {\displaystyle {\tfrac {1}{2}}x^{2}}
2 x {\displaystyle 2x} {\displaystyle 2x} x 2 {\displaystyle x^{2}} {\displaystyle x^{2}}
x 2 {\displaystyle x^{2}} {\displaystyle x^{2}} 1 3 x 3 {\displaystyle {\tfrac {1}{3}}x^{3}} {\displaystyle {\tfrac {1}{3}}x^{3}}
3 x 2 {\displaystyle 3x^{2}} {\displaystyle 3x^{2}} x 3 {\displaystyle x^{3}} {\displaystyle x^{3}}
x {\displaystyle {\sqrt {x}}} {\displaystyle {\sqrt {x}}} 2 3 x 3 2 {\displaystyle {\tfrac {2}{3}}x^{\tfrac {3}{2}}} {\displaystyle {\tfrac {2}{3}}x^{\tfrac {3}{2}}}
x n {\displaystyle {\sqrt[{n}]{x}}} {\displaystyle {\sqrt[{n}]{x}}} n n + 1 ( x n ) n + 1 ( n ≠ − 1 ) {\displaystyle {\frac {n}{n+1}}({\sqrt[{n}]{x}})^{n+1}\quad (n\neq -1)} {\displaystyle {\frac {n}{n+1}}({\sqrt[{n}]{x}})^{n+1}\quad (n\neq -1)}
1 x {\displaystyle {\frac {1}{\sqrt {x}}}} {\displaystyle {\frac {1}{\sqrt {x}}}} 2 x {\displaystyle 2{\sqrt {x}}} {\displaystyle 2{\sqrt {x}}}
1 n ( x n − 1 n ) {\displaystyle {\frac {1}{n({\sqrt[{n}]{x^{n-1}}})}}} {\displaystyle {\frac {1}{n({\sqrt[{n}]{x^{n-1}}})}}} x n {\displaystyle {\sqrt[{n}]{x}}} {\displaystyle {\sqrt[{n}]{x}}}
− 2 x 3 {\displaystyle -{\frac {2}{x^{3}}}} {\displaystyle -{\frac {2}{x^{3}}}} 1 x 2 {\displaystyle {\frac {1}{x^{2}}}} {\displaystyle {\frac {1}{x^{2}}}}
− 1 x 2 {\displaystyle -{\frac {1}{x^{2}}}} {\displaystyle -{\frac {1}{x^{2}}}} 1 x {\displaystyle {\frac {1}{x}}} {\displaystyle {\frac {1}{x}}}

Exponential- und Logarithmusfunktionen

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Funktion f ( x ) {\displaystyle f(x)} {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} {\displaystyle F(x)}
e x {\displaystyle \mathrm {e} ^{x}} {\displaystyle \mathrm {e} ^{x}} e x {\displaystyle \mathrm {e} ^{x}} {\displaystyle \mathrm {e} ^{x}}
e k x {\displaystyle \mathrm {e} ^{kx}} {\displaystyle \mathrm {e} ^{kx}} 1 k e k x {\displaystyle {\frac {1}{k}}\mathrm {e} ^{kx}} {\displaystyle {\frac {1}{k}}\mathrm {e} ^{kx}}
a x ln ⁡ a ( a > 0 ) {\displaystyle a^{x}\ln a\quad (a>0)} {\displaystyle a^{x}\ln a\quad (a>0)} a x {\displaystyle a^{x}} {\displaystyle a^{x}}
a x {\displaystyle a^{x}} {\displaystyle a^{x}} a x ln ⁡ a {\displaystyle {\frac {a^{x}}{\ln a}}} {\displaystyle {\frac {a^{x}}{\ln a}}}
x x ( 1 + ln ⁡ ( x ) ) {\displaystyle x^{x}(1+\ln(x))} {\displaystyle x^{x}(1+\ln(x))} x x ( x > 0 ) {\displaystyle x^{x}\quad (x>0)} {\displaystyle x^{x}\quad (x>0)}
e x ln ⁡ | x | ( ln ⁡ | x | + 1 ) {\displaystyle \mathrm {e} ^{x\ln \left|x\right|}(\ln \left|x\right|+1)} {\displaystyle \mathrm {e} ^{x\ln \left|x\right|}(\ln \left|x\right|+1)} | x | x = e x ln ⁡ | x | ( x ≠ 0 ) {\displaystyle \left|x\right|^{x}=\mathrm {e} ^{x\ln \left|x\right|}\quad (x\neq 0)} {\displaystyle \left|x\right|^{x}=\mathrm {e} ^{x\ln \left|x\right|}\quad (x\neq 0)}
1 x {\displaystyle {\frac {1}{x}}} {\displaystyle {\frac {1}{x}}} ln ⁡ | x | {\displaystyle \ln \left|x\right|} {\displaystyle \ln \left|x\right|}[A 1]
ln ⁡ x {\displaystyle \ln x} {\displaystyle \ln x} x ln ⁡ ( x ) − x {\displaystyle x\ln(x)-x} {\displaystyle x\ln(x)-x}
x n ln ⁡ x {\displaystyle x^{n}\ln x} {\displaystyle x^{n}\ln x} x n + 1 n + 1 ( ln ⁡ x − 1 n + 1 ) ( n ≥ 0 ) {\displaystyle {\frac {x^{n+1}}{n+1}}\left(\ln x-{\frac {1}{n+1}}\right)\quad (n\geq 0)} {\displaystyle {\frac {x^{n+1}}{n+1}}\left(\ln x-{\frac {1}{n+1}}\right)\quad (n\geq 0)}
u ′ ( x ) ln ⁡ u ( x ) {\displaystyle u'(x)\ln u(x)} {\displaystyle u'(x)\ln u(x)} u ( x ) ln ⁡ u ( x ) − u ( x ) {\displaystyle u(x)\ln u(x)-u(x)} {\displaystyle u(x)\ln u(x)-u(x)}
1 x ln n ⁡ x ( n ≠ − 1 ) {\displaystyle {\frac {1}{x}}\ln ^{n}x\quad (n\neq -1)} {\displaystyle {\frac {1}{x}}\ln ^{n}x\quad (n\neq -1)} 1 n + 1 ln n + 1 ⁡ x {\displaystyle {\frac {1}{n+1}}\ln ^{n+1}x} {\displaystyle {\frac {1}{n+1}}\ln ^{n+1}x}
1 x ln ⁡ x n ( n ≠ 0 ) {\displaystyle {\frac {1}{x}}\ln {x^{n}}\quad (n\neq 0)} {\displaystyle {\frac {1}{x}}\ln {x^{n}}\quad (n\neq 0)} 1 2 n ln 2 ⁡ x n = n 2 ln 2 ⁡ x {\displaystyle {\frac {1}{2n}}\ln ^{2}{x^{n}}={\frac {n}{2}}\ln ^{2}x} {\displaystyle {\frac {1}{2n}}\ln ^{2}{x^{n}}={\frac {n}{2}}\ln ^{2}x}
1 x 1 ln ⁡ a {\displaystyle {\frac {1}{x}}{\frac {1}{\ln a}}} {\displaystyle {\frac {1}{x}}{\frac {1}{\ln a}}} log a ⁡ x {\displaystyle \log _{a}x} {\displaystyle \log _{a}x}
1 x ln ⁡ x {\displaystyle {\frac {1}{x\ln x}}} {\displaystyle {\frac {1}{x\ln x}}} ln ⁡ | ln ⁡ x | ( x > 0 , x ≠ 1 ) {\displaystyle \ln \left|\ln x\right|\quad (x>0,x\neq 1)} {\displaystyle \ln \left|\ln x\right|\quad (x>0,x\neq 1)}
log a ⁡ x {\displaystyle \log _{a}x} {\displaystyle \log _{a}x} 1 ln ⁡ a ( x ln ⁡ x − x ) {\displaystyle {\frac {1}{\ln a}}(x\ln x-x)} {\displaystyle {\frac {1}{\ln a}}(x\ln x-x)}

Anmerkung:

  1. ↑ Sonderfall von x n {\displaystyle x^{n}} {\displaystyle x^{n}} für n = − 1 {\displaystyle n=-1} {\displaystyle n=-1}, siehe oben in „Potenz- und Wurzelfunktionen"

Trigonometrische Funktionen und Hyperbelfunktionen

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Trigonometrische Funktionen

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Funktion f ( x ) {\displaystyle f(x)} {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} {\displaystyle F(x)}
sin ⁡ ( x ) {\displaystyle \sin(x)} {\displaystyle \sin(x)} − cos ⁡ ( x ) {\displaystyle -\cos(x)} {\displaystyle -\cos(x)}
cos ⁡ ( x ) {\displaystyle \cos(x)} {\displaystyle \cos(x)} sin ⁡ ( x ) {\displaystyle \sin(x)} {\displaystyle \sin(x)}
tan ⁡ ( x ) = sin ⁡ ( x ) / cos ⁡ ( x ) {\displaystyle \tan(x)=\sin(x)/\cos(x)} {\displaystyle \tan(x)=\sin(x)/\cos(x)} ln ⁡ [ sec ⁡ ( x ) ] {\displaystyle \ln {\bigl [}\sec(x){\bigr ]}} {\displaystyle \ln {\bigl [}\sec(x){\bigr ]}}
cot ⁡ ( x ) = cos ⁡ ( x ) / sin ⁡ ( x ) {\displaystyle \cot(x)=\cos(x)/\sin(x)} {\displaystyle \cot(x)=\cos(x)/\sin(x)} − ln ⁡ [ csc ⁡ ( x ) ] {\displaystyle -\ln {\bigl [}\csc(x){\bigr ]}} {\displaystyle -\ln {\bigl [}\csc(x){\bigr ]}}
sec ⁡ ( x ) = 1 / cos ⁡ ( x ) {\displaystyle \sec(x)=1/\cos(x)} {\displaystyle \sec(x)=1/\cos(x)} artanh ⁡ [ sin ⁡ ( x ) ] {\displaystyle \operatorname {artanh} {\bigl [}\sin(x){\bigr ]}} {\displaystyle \operatorname {artanh} {\bigl [}\sin(x){\bigr ]}}
csc ⁡ ( x ) = 1 / sin ⁡ ( x ) {\displaystyle \csc(x)=1/\sin(x)} {\displaystyle \csc(x)=1/\sin(x)} − artanh ⁡ [ cos ⁡ ( x ) ] {\displaystyle -\operatorname {artanh} {\bigl [}\cos(x){\bigr ]}} {\displaystyle -\operatorname {artanh} {\bigl [}\cos(x){\bigr ]}}
sec 2 ⁡ ( x ) = 1 + tan 2 ⁡ ( x ) {\displaystyle \operatorname {sec} ^{2}(x)=1+\tan ^{2}(x)} {\displaystyle \operatorname {sec} ^{2}(x)=1+\tan ^{2}(x)} tan ⁡ ( x ) {\displaystyle \tan(x)} {\displaystyle \tan(x)}
− csc 2 ⁡ ( x ) = − [ 1 + cot 2 ⁡ ( x ) ] {\displaystyle -\operatorname {csc} ^{2}(x)=-{\bigl [}1+\cot ^{2}(x){\bigr ]}} {\displaystyle -\operatorname {csc} ^{2}(x)=-{\bigl [}1+\cot ^{2}(x){\bigr ]}} cot ⁡ ( x ) {\displaystyle \cot(x)} {\displaystyle \cot(x)}
sin 2 ⁡ ( x ) {\displaystyle \sin ^{2}(x)} {\displaystyle \sin ^{2}(x)} 1 2 [ x − sin ⁡ ( x ) cos ⁡ ( x ) ] = 1 2 x − 1 4 sin ⁡ ( 2 x ) {\displaystyle {\tfrac {1}{2}}{\bigl [}x-\sin(x)\cos(x){\bigr ]}={\tfrac {1}{2}}x-{\tfrac {1}{4}}\sin(2x)} {\displaystyle {\tfrac {1}{2}}{\bigl [}x-\sin(x)\cos(x){\bigr ]}={\tfrac {1}{2}}x-{\tfrac {1}{4}}\sin(2x)}
cos 2 ⁡ ( x ) {\displaystyle \cos ^{2}(x)} {\displaystyle \cos ^{2}(x)} 1 2 [ x + sin ⁡ ( x ) cos ⁡ ( x ) ] = 1 2 x + 1 4 sin ⁡ ( 2 x ) {\displaystyle {\tfrac {1}{2}}{\bigl [}x+\sin(x)\cos(x){\bigr ]}={\tfrac {1}{2}}x+{\tfrac {1}{4}}\sin(2x)} {\displaystyle {\tfrac {1}{2}}{\bigl [}x+\sin(x)\cos(x){\bigr ]}={\tfrac {1}{2}}x+{\tfrac {1}{4}}\sin(2x)}
sin ⁡ ( k x ) ⋅ cos ⁡ ( k x ) {\displaystyle \sin(kx)\cdot \cos(kx)} {\displaystyle \sin(kx)\cdot \cos(kx)} − 1 4 k cos ⁡ ( 2 k x ) {\displaystyle -{\frac {1}{4k}}\cos(2kx)} {\displaystyle -{\frac {1}{4k}}\cos(2kx)}
sin ⁡ ( k x ) ⋅ cos ⁡ ( k x ) {\displaystyle \sin(kx)\cdot \cos(kx)} {\displaystyle \sin(kx)\cdot \cos(kx)} 1 2 k sin 2 ⁡ ( k x ) {\displaystyle {\frac {1}{2k}}\sin ^{2}(kx)} {\displaystyle {\frac {1}{2k}}\sin ^{2}(kx)}
sin ⁡ ( a x ) exp ⁡ ( b x ) {\displaystyle {\frac {\sin(ax)}{\exp(bx)}}} {\displaystyle {\frac {\sin(ax)}{\exp(bx)}}} a exp ⁡ ( b x ) − a cos ⁡ ( a x ) − b sin ⁡ ( a x ) ( a 2 + b 2 ) exp ⁡ ( b x ) {\displaystyle {\frac {a\exp(bx)-a\cos(ax)-b\sin(ax)}{(a^{2}+b^{2})\exp(bx)}}} {\displaystyle {\frac {a\exp(bx)-a\cos(ax)-b\sin(ax)}{(a^{2}+b^{2})\exp(bx)}}}
cos ⁡ ( a x ) exp ⁡ ( b x ) {\displaystyle {\frac {\cos(ax)}{\exp(bx)}}} {\displaystyle {\frac {\cos(ax)}{\exp(bx)}}} a sin ⁡ ( a x ) − b cos ⁡ ( a x ) + b exp ⁡ ( b x ) ( a 2 + b 2 ) exp ⁡ ( b x ) {\displaystyle {\frac {a\sin(ax)-b\cos(ax)+b\exp(bx)}{(a^{2}+b^{2})\exp(bx)}}} {\displaystyle {\frac {a\sin(ax)-b\cos(ax)+b\exp(bx)}{(a^{2}+b^{2})\exp(bx)}}}
arcsin ⁡ x {\displaystyle \arcsin x} {\displaystyle \arcsin x} x arcsin ⁡ x + 1 − x 2 {\displaystyle x\arcsin x+{\sqrt {1-x^{2}}}} {\displaystyle x\arcsin x+{\sqrt {1-x^{2}}}}
arccos ⁡ x {\displaystyle \arccos x} {\displaystyle \arccos x} x arccos ⁡ x − 1 − x 2 {\displaystyle x\arccos x-{\sqrt {1-x^{2}}}} {\displaystyle x\arccos x-{\sqrt {1-x^{2}}}}
arctan ⁡ x {\displaystyle \arctan x} {\displaystyle \arctan x} x arctan ⁡ x − 1 2 ln ⁡ ( 1 + x 2 ) {\displaystyle x\arctan x-{\tfrac {1}{2}}\ln \left(1+x^{2}\right)} {\displaystyle x\arctan x-{\tfrac {1}{2}}\ln \left(1+x^{2}\right)}
arccot ⁡ x {\displaystyle \operatorname {arccot} x} {\displaystyle \operatorname {arccot} x} x arccot ⁡ x + 1 2 ln ⁡ ( 1 + x 2 ) {\displaystyle x\operatorname {arccot} x+{\tfrac {1}{2}}\ln \left(1+x^{2}\right)} {\displaystyle x\operatorname {arccot} x+{\tfrac {1}{2}}\ln \left(1+x^{2}\right)}
1 1 − x 2 {\displaystyle {\frac {1}{\sqrt {1-x^{2}}}}} {\displaystyle {\frac {1}{\sqrt {1-x^{2}}}}} arcsin ⁡ x {\displaystyle \arcsin x} {\displaystyle \arcsin x}
− 1 1 − x 2 {\displaystyle {\frac {-1}{\sqrt {1-x^{2}}}}} {\displaystyle {\frac {-1}{\sqrt {1-x^{2}}}}} arccos ⁡ x {\displaystyle \arccos x} {\displaystyle \arccos x}
1 x 2 + 1 {\displaystyle {\frac {1}{x^{2}+1}}} {\displaystyle {\frac {1}{x^{2}+1}}} arctan ⁡ x {\displaystyle \arctan x} {\displaystyle \arctan x}
− 1 x 2 + 1 {\displaystyle -{\frac {1}{x^{2}+1}}} {\displaystyle -{\frac {1}{x^{2}+1}}} arccot ⁡ x {\displaystyle \operatorname {arccot} x} {\displaystyle \operatorname {arccot} x}
x 2 x 2 + 1 {\displaystyle {\frac {x^{2}}{x^{2}+1}}} {\displaystyle {\frac {x^{2}}{x^{2}+1}}} x − arctan ⁡ x {\displaystyle x-\arctan x} {\displaystyle x-\arctan x}
1 ( x 2 + 1 ) 2 {\displaystyle {\frac {1}{(x^{2}+1)^{2}}}} {\displaystyle {\frac {1}{(x^{2}+1)^{2}}}} 1 2 ( x x 2 + 1 + arctan ⁡ x ) {\displaystyle {\frac {1}{2}}\left({\frac {x}{x^{2}+1}}+\arctan x\right)} {\displaystyle {\frac {1}{2}}\left({\frac {x}{x^{2}+1}}+\arctan x\right)}
a 2 − x 2 {\displaystyle {\sqrt {a^{2}-x^{2}}}} {\displaystyle {\sqrt {a^{2}-x^{2}}}} a 2 2 arcsin ⁡ ( x a ) + x 2 a 2 − x 2 {\displaystyle {\frac {a^{2}}{2}}\arcsin \left({\frac {x}{a}}\right)+{\frac {x}{2}}{\sqrt {a^{2}-x^{2}}}} {\displaystyle {\frac {a^{2}}{2}}\arcsin \left({\frac {x}{a}}\right)+{\frac {x}{2}}{\sqrt {a^{2}-x^{2}}}}
1 a x 2 + b x + c {\displaystyle {\frac {1}{ax^{2}+bx+c}}} {\displaystyle {\frac {1}{ax^{2}+bx+c}}} 2 4 a c − b 2 arctan ⁡ ( 2 a x + b 4 a c − b 2 ) {\displaystyle {\frac {2}{\sqrt {4ac-b^{2}}}}\arctan {\biggl (}{\frac {2ax+b}{\sqrt {4ac-b^{2}}}}{\biggr )}} {\displaystyle {\frac {2}{\sqrt {4ac-b^{2}}}}\arctan {\biggl (}{\frac {2ax+b}{\sqrt {4ac-b^{2}}}}{\biggr )}}

Hyperbelfunktionen

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Funktion f ( x ) {\displaystyle f(x)} {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} {\displaystyle F(x)}
sinh ⁡ ( x ) {\displaystyle \sinh(x)} {\displaystyle \sinh(x)} cosh ⁡ ( x ) {\displaystyle \cosh(x)} {\displaystyle \cosh(x)}
cosh ⁡ ( x ) {\displaystyle \cosh(x)} {\displaystyle \cosh(x)} sinh ⁡ ( x ) {\displaystyle \sinh(x)} {\displaystyle \sinh(x)}
tanh ⁡ ( x ) {\displaystyle \tanh(x)} {\displaystyle \tanh(x)} ln ⁡ [ cosh ⁡ ( x ) ] {\displaystyle \ln {\bigl [}\cosh(x){\bigr ]}} {\displaystyle \ln {\bigl [}\cosh(x){\bigr ]}}
coth ⁡ ( x ) {\displaystyle \coth(x)} {\displaystyle \coth(x)} ln ⁡ | sinh ⁡ ( x ) | {\displaystyle \ln |{\sinh(x)}|} {\displaystyle \ln |{\sinh(x)}|}
sech ⁡ ( x ) {\displaystyle \operatorname {sech} (x)} {\displaystyle \operatorname {sech} (x)} gd ⁡ ( x ) = arctan ⁡ [ sinh ⁡ ( x ) ] {\displaystyle \operatorname {gd} (x)=\arctan {\bigl [}\sinh(x){\bigr ]}} {\displaystyle \operatorname {gd} (x)=\arctan {\bigl [}\sinh(x){\bigr ]}}
csch ⁡ ( x ) {\displaystyle \operatorname {csch} (x)} {\displaystyle \operatorname {csch} (x)} − arcoth ⁡ [ cosh ⁡ ( x ) ] {\displaystyle -\operatorname {arcoth} {\bigl [}\cosh(x){\bigr ]}} {\displaystyle -\operatorname {arcoth} {\bigl [}\cosh(x){\bigr ]}}
1 cosh 2 ⁡ x = 1 − tanh 2 ⁡ x {\displaystyle {\frac {1}{\cosh ^{2}x}}=1-\tanh ^{2}x} {\displaystyle {\frac {1}{\cosh ^{2}x}}=1-\tanh ^{2}x} tanh ⁡ x {\displaystyle \tanh x} {\displaystyle \tanh x}
− 1 sinh 2 ⁡ x = 1 − coth 2 ⁡ x {\displaystyle {\frac {-1}{\sinh ^{2}x}}=1-\coth ^{2}x} {\displaystyle {\frac {-1}{\sinh ^{2}x}}=1-\coth ^{2}x} coth ⁡ x {\displaystyle \coth x} {\displaystyle \coth x}
arsinh ⁡ x {\displaystyle \operatorname {arsinh} x} {\displaystyle \operatorname {arsinh} x} x arsinh ⁡ x − x 2 + 1 {\displaystyle x\operatorname {arsinh} x-{\sqrt {x^{2}+1}}} {\displaystyle x\operatorname {arsinh} x-{\sqrt {x^{2}+1}}}
arcosh ⁡ x {\displaystyle \operatorname {arcosh} x} {\displaystyle \operatorname {arcosh} x} x arcosh ⁡ x − x 2 − 1 {\displaystyle x\operatorname {arcosh} x-{\sqrt {x^{2}-1}}} {\displaystyle x\operatorname {arcosh} x-{\sqrt {x^{2}-1}}}
artanh ⁡ x {\displaystyle \operatorname {artanh} x} {\displaystyle \operatorname {artanh} x} x artanh ⁡ x + 1 2 ln ⁡ ( 1 − x 2 ) {\displaystyle x\operatorname {artanh} x+{\frac {1}{2}}\ln {\left(1-x^{2}\right)}} {\displaystyle x\operatorname {artanh} x+{\frac {1}{2}}\ln {\left(1-x^{2}\right)}}
arcoth ⁡ x {\displaystyle \operatorname {arcoth} x} {\displaystyle \operatorname {arcoth} x} x arcoth ⁡ x + 1 2 ln ⁡ ( x 2 − 1 ) {\displaystyle x\operatorname {arcoth} x+{\frac {1}{2}}\ln {\left(x^{2}-1\right)}} {\displaystyle x\operatorname {arcoth} x+{\frac {1}{2}}\ln {\left(x^{2}-1\right)}}
1 x 2 + 1 {\displaystyle {\frac {1}{\sqrt {x^{2}+1}}}} {\displaystyle {\frac {1}{\sqrt {x^{2}+1}}}} arsinh ⁡ x {\displaystyle \operatorname {arsinh} x} {\displaystyle \operatorname {arsinh} x}
1 x 2 − 1 ( x > 1 ) {\displaystyle {\frac {1}{\sqrt {x^{2}-1}}}\quad (x>1)} {\displaystyle {\frac {1}{\sqrt {x^{2}-1}}}\quad (x>1)} arcosh ⁡ x {\displaystyle \operatorname {arcosh} x} {\displaystyle \operatorname {arcosh} x}
a 2 + x 2 {\displaystyle {\sqrt {a^{2}+x^{2}}}} {\displaystyle {\sqrt {a^{2}+x^{2}}}} a 2 2 arsinh ⁡ ( x a ) + x 2 a 2 + x 2 {\displaystyle {\frac {a^{2}}{2}}\operatorname {arsinh} \left({\frac {x}{a}}\right)+{\frac {x}{2}}{\sqrt {a^{2}+x^{2}}}} {\displaystyle {\frac {a^{2}}{2}}\operatorname {arsinh} \left({\frac {x}{a}}\right)+{\frac {x}{2}}{\sqrt {a^{2}+x^{2}}}}
1 a x 2 + b x + c {\displaystyle {\frac {1}{\sqrt {ax^{2}+bx+c}}}} {\displaystyle {\frac {1}{\sqrt {ax^{2}+bx+c}}}} 1 a arsinh ⁡ ( 2 a x + b 4 a c − b 2 ) {\displaystyle {\frac {1}{\sqrt {a}}}\operatorname {arsinh} {\biggl (}{\frac {2ax+b}{\sqrt {4ac-b^{2}}}}{\biggr )}} {\displaystyle {\frac {1}{\sqrt {a}}}\operatorname {arsinh} {\biggl (}{\frac {2ax+b}{\sqrt {4ac-b^{2}}}}{\biggr )}}
1 1 − x 2 ( | x | < 1 ) {\displaystyle {\frac {1}{1-x^{2}}}\quad (\left|x\right|<1)} {\displaystyle {\frac {1}{1-x^{2}}}\quad (\left|x\right|<1)} artanh ⁡ x {\displaystyle \operatorname {artanh} x} {\displaystyle \operatorname {artanh} x}
1 1 − x 2 ( | x | > 1 ) {\displaystyle {\frac {1}{1-x^{2}}}\quad (\left|x\right|>1)} {\displaystyle {\frac {1}{1-x^{2}}}\quad (\left|x\right|>1)} arcoth ⁡ x {\displaystyle \operatorname {arcoth} x} {\displaystyle \operatorname {arcoth} x}

Elliptische Funktionen und elliptische Integrale

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Viele Stammfunktionen von algebraischen Funktionen können nicht elementar dargestellt werden. Für die Darstellung von den Stammfunktionen dieser algebraischen Funktionen genügen für die Darstellung nicht die Kreisbogenmaßfunktionen, die Hyperbelflächenmaßfunktionen, die Logarithmen und die algebraischen Funktionen alleine. Diese nicht elementar darstellbaren Integrale von den genannten algebraischen Funktionen werden elliptische Integrale genannt. Ihre Umkehrfunktionen werden als elliptische Funktionen bezeichnet. Diejenigen elliptischen Integrale, welche den Definitionsbereich der betroffenen algebraischen Funktion komplett abschließen, werden vollständige elliptische Integrale genannt. Der Quotient des vollständigen elliptischen Integrals erster Art vom Pythagoräisch komplementären Modul dividiert durch das vollständige elliptische Integral erster Art vom betroffenen Modul selbst wird als reelles Halbperiodenverhältnis oder als reelles Periodenverhältnis bezeichnet. Das elliptische Nomen ist die Exponentialfunktion aus dem negativen Produkt der Kreiszahl und des reellen Periodenverhältnisses. Die Jacobischen Thetafunktionen ordnen das elliptische Nomen den algebraischen Vielfachen von der Quadratwurzel des vollständigen elliptischen Integrals erster Art zu. Ebenso werden diejenigen Funktionen als elliptische Funktionen bezeichnet, welche als algebraische Kombinationen aus den Jacobischen Thetafunktionen hervorgehen.

Elliptische Stammfunktionen von algebraischen Wurzelfunktionen

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Funktion f ( x ) {\displaystyle f(x)} {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} {\displaystyle F(x)}
[ 1 − k 2 sin ⁡ ( x ) 2 ] − 1 / 2 {\displaystyle {\bigl [}1-k^{2}\sin(x)^{2}{\bigr ]}^{-1/2}} {\displaystyle {\bigl [}1-k^{2}\sin(x)^{2}{\bigr ]}^{-1/2}} F ( x ; k ) {\displaystyle F(x;k)} {\displaystyle F(x;k)}
[ 1 − k 2 sin ⁡ ( x ) 2 ] 1 / 2 {\displaystyle {\bigl [}1-k^{2}\sin(x)^{2}{\bigr ]}^{1/2}} {\displaystyle {\bigl [}1-k^{2}\sin(x)^{2}{\bigr ]}^{1/2}} E ( x ; k ) {\displaystyle E(x;k)} {\displaystyle E(x;k)}
1 1 − x 4 {\displaystyle {\frac {1}{\sqrt {1-x^{4}}}}} {\displaystyle {\frac {1}{\sqrt {1-x^{4}}}}} arcsl ⁡ ( x ) = 1 2 2 K ( 1 2 2 ) − 1 2 2 F [ arccos ⁡ ( x ) ; 1 2 2 ] {\displaystyle \operatorname {arcsl} (x)={\frac {1}{2}}{\sqrt {2}},円K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-{\frac {1}{2}}{\sqrt {2}},円F{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}} {\displaystyle \operatorname {arcsl} (x)={\frac {1}{2}}{\sqrt {2}},円K{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-{\frac {1}{2}}{\sqrt {2}},円F{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}}
x 2 + 1 1 − x 4 {\displaystyle {\frac {x^{2}+1}{\sqrt {1-x^{4}}}}} {\displaystyle {\frac {x^{2}+1}{\sqrt {1-x^{4}}}}} 2 E ( 1 2 2 ) − 2 E [ arccos ⁡ ( x ) ; 1 2 2 ] {\displaystyle {\sqrt {2}},円E{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-{\sqrt {2}},円E{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}} {\displaystyle {\sqrt {2}},円E{\bigl (}{\frac {1}{2}}{\sqrt {2}}{\bigr )}-{\sqrt {2}},円E{\biggl [}\arccos(x);{\frac {1}{2}}{\sqrt {2}}{\biggr ]}}
x 2 1 − x 4 {\displaystyle {\frac {x^{2}}{\sqrt {1-x^{4}}}}} {\displaystyle {\frac {x^{2}}{\sqrt {1-x^{4}}}}} 1 arcsl ⁡ ( x ) ∫ 0 1 y 2 + 1 2 y 2 [ artanh ( y 2 ) − artanh ( 1 − x 4 y 2 1 − x 4 y 4 ) ] d y {\displaystyle {\frac {1}{\operatorname {arcsl} (x)}}\int _{0}^{1}{\frac {y^{2}+1}{2y^{2}}}{\biggl [}{\text{artanh}}{\bigl (}y^{2}{\bigr )}-{\text{artanh}}{\biggl (}{\frac {{\sqrt {1-x^{4}}},円y^{2}}{\sqrt {1-x^{4}y^{4}}}}{\biggr )}{\biggr ]}\mathrm {d} y} {\displaystyle {\frac {1}{\operatorname {arcsl} (x)}}\int _{0}^{1}{\frac {y^{2}+1}{2y^{2}}}{\biggl [}{\text{artanh}}{\bigl (}y^{2}{\bigr )}-{\text{artanh}}{\biggl (}{\frac {{\sqrt {1-x^{4}}},円y^{2}}{\sqrt {1-x^{4}y^{4}}}}{\biggr )}{\biggr ]}\mathrm {d} y}
1 x 4 + 1 {\displaystyle {\frac {1}{\sqrt {x^{4}+1}}}} {\displaystyle {\frac {1}{\sqrt {x^{4}+1}}}} 2 arcsl ⁡ [ x ( x 4 + 1 + 1 ) − 1 / 2 ] {\displaystyle {\sqrt {2}},円\operatorname {arcsl} \left[x({\sqrt {x^{4}+1}}+1)^{-1/2}\right]} {\displaystyle {\sqrt {2}},円\operatorname {arcsl} \left[x({\sqrt {x^{4}+1}}+1)^{-1/2}\right]}
1 1 − x 6 {\displaystyle {\frac {1}{\sqrt {1-x^{6}}}}} {\displaystyle {\frac {1}{\sqrt {1-x^{6}}}}} 1 6 27 4 F [ 2 arctan ⁡ ( 3 4 x 1 − x 2 ) ; sin ⁡ ( π 12 ) ] {\displaystyle {\frac {1}{6}}{\sqrt[{4}]{27}},円F{\biggl [}2\arctan {\biggl (}{\frac {{\sqrt[{4}]{3}},円x}{\sqrt {1-x^{2}}}}{\biggr )};\sin {\bigl (}{\frac {\pi }{12}}{\bigr )}{\biggr ]}} {\displaystyle {\frac {1}{6}}{\sqrt[{4}]{27}},円F{\biggl [}2\arctan {\biggl (}{\frac {{\sqrt[{4}]{3}},円x}{\sqrt {1-x^{2}}}}{\biggr )};\sin {\bigl (}{\frac {\pi }{12}}{\bigr )}{\biggr ]}}
1 x 6 + 1 {\displaystyle {\frac {1}{\sqrt {x^{6}+1}}}} {\displaystyle {\frac {1}{\sqrt {x^{6}+1}}}} 1 6 27 4 F [ 2 arctan ⁡ ( 3 4 x x 2 + 1 ) ; cos ⁡ ( π 12 ) ] {\displaystyle {\frac {1}{6}}{\sqrt[{4}]{27}},円F{\biggl [}2\arctan {\biggl (}{\frac {{\sqrt[{4}]{3}},円x}{\sqrt {x^{2}+1}}}{\biggr )};\cos {\bigl (}{\frac {\pi }{12}}{\bigr )}{\biggr ]}} {\displaystyle {\frac {1}{6}}{\sqrt[{4}]{27}},円F{\biggl [}2\arctan {\biggl (}{\frac {{\sqrt[{4}]{3}},円x}{\sqrt {x^{2}+1}}}{\biggr )};\cos {\bigl (}{\frac {\pi }{12}}{\bigr )}{\biggr ]}}
1 − x 2 x 8 + 1 {\displaystyle {\frac {1-x^{2}}{\sqrt {x^{8}+1}}}} {\displaystyle {\frac {1-x^{2}}{\sqrt {x^{8}+1}}}} 1 2 sec ⁡ ( π 8 ) F { arcsin ⁡ [ 2 cos ⁡ ( π / 8 ) x x 2 + 1 ] ; tan ⁡ ( π 8 ) } {\displaystyle {\frac {1}{2}}\sec \left({\frac {\pi }{8}}\right)F\left\{\arcsin \left[{\frac {2\cos(\pi /8)x}{x^{2}+1}}\right];\tan \left({\frac {\pi }{8}}\right)\right\}} {\displaystyle {\frac {1}{2}}\sec \left({\frac {\pi }{8}}\right)F\left\{\arcsin \left[{\frac {2\cos(\pi /8)x}{x^{2}+1}}\right];\tan \left({\frac {\pi }{8}}\right)\right\}}
x 2 + 1 x 8 + 1 {\displaystyle {\frac {x^{2}+1}{\sqrt {x^{8}+1}}}} {\displaystyle {\frac {x^{2}+1}{\sqrt {x^{8}+1}}}} 1 2 sec ⁡ ( π 8 ) F { 2 arctan ⁡ [ 2 cos ⁡ ( π / 8 ) x x 4 + 2 x 2 + 1 − x 2 + 1 ] ; 2 2 4 sin ⁡ ( π 8 ) } {\displaystyle {\frac {1}{2}}\sec \left({\frac {\pi }{8}}\right)F\left\{2\arctan \left[{\frac {2\cos(\pi /8)x}{{\sqrt {x^{4}+{\sqrt {2}}x^{2}+1}}-x^{2}+1}}\right];2{\sqrt[{4}]{2}}\sin \left({\frac {\pi }{8}}\right)\right\}} {\displaystyle {\frac {1}{2}}\sec \left({\frac {\pi }{8}}\right)F\left\{2\arctan \left[{\frac {2\cos(\pi /8)x}{{\sqrt {x^{4}+{\sqrt {2}}x^{2}+1}}-x^{2}+1}}\right];2{\sqrt[{4}]{2}}\sin \left({\frac {\pi }{8}}\right)\right\}}
1 − ( 2 + 1 ) x 2 1 − x 8 {\displaystyle {\frac {1-({\sqrt {2}}+1),円x^{2}}{\sqrt {1-x^{8}}}}} {\displaystyle {\frac {1-({\sqrt {2}}+1),円x^{2}}{\sqrt {1-x^{8}}}}} F [ arcsin ⁡ ( x 1 − x 2 1 + x 2 ) ; tan ⁡ ( π 8 ) ] {\displaystyle F{\biggl [}\arcsin {\biggl (}{\frac {x{\sqrt {1-x^{2}}}}{\sqrt {1+x^{2}}}}{\biggr )};\tan {\bigl (}{\frac {\pi }{8}}{\bigr )}{\biggr ]}} {\displaystyle F{\biggl [}\arcsin {\biggl (}{\frac {x{\sqrt {1-x^{2}}}}{\sqrt {1+x^{2}}}}{\biggr )};\tan {\bigl (}{\frac {\pi }{8}}{\bigr )}{\biggr ]}}
( 2 + 1 ) x 2 + 1 1 − x 8 {\displaystyle {\frac {({\sqrt {2}}+1),円x^{2}+1}{\sqrt {1-x^{8}}}}} {\displaystyle {\frac {({\sqrt {2}}+1),円x^{2}+1}{\sqrt {1-x^{8}}}}} F [ arctan ⁡ ( x 1 + x 2 1 − x 2 ) ; 2 2 4 sin ⁡ ( π 8 ) ] {\displaystyle F{\biggl [}\arctan {\biggl (}{\frac {x{\sqrt {1+x^{2}}}}{\sqrt {1-x^{2}}}}{\biggr )};2{\sqrt[{4}]{2}}\sin {\bigl (}{\frac {\pi }{8}}{\bigr )}{\biggr ]}} {\displaystyle F{\biggl [}\arctan {\biggl (}{\frac {x{\sqrt {1+x^{2}}}}{\sqrt {1-x^{2}}}}{\biggr )};2{\sqrt[{4}]{2}}\sin {\bigl (}{\frac {\pi }{8}}{\bigr )}{\biggr ]}}
1 ( a x 2 + b x + c ) 3 4 {\displaystyle {\frac {1}{\sqrt[{4}]{(ax^{2}+bx+c)^{3}}}}} {\displaystyle {\frac {1}{\sqrt[{4}]{(ax^{2}+bx+c)^{3}}}}} 2 2 4 a 2 c − a b 2 4 arcsl ⁡ [ 2 a x + b 4 a ( a x 2 + b x + c ) + 4 a c − b 2 ] {\displaystyle {\frac {2{\sqrt {2}}}{\sqrt[{4}]{4a^{2}c-ab^{2}}}}\operatorname {arcsl} \left[{\frac {2ax+b}{{\sqrt {4a(ax^{2}+bx+c)}}+{\sqrt {4ac-b^{2}}}}}\right]} {\displaystyle {\frac {2{\sqrt {2}}}{\sqrt[{4}]{4a^{2}c-ab^{2}}}}\operatorname {arcsl} \left[{\frac {2ax+b}{{\sqrt {4a(ax^{2}+bx+c)}}+{\sqrt {4ac-b^{2}}}}}\right]}
1 ( x 2 + 2 v x + 1 ) ( x 2 + 2 w x + 1 ) {\displaystyle {\frac {1}{\sqrt {(x^{2}+2vx+1)(x^{2}+2wx+1)}}}} {\displaystyle {\frac {1}{\sqrt {(x^{2}+2vx+1)(x^{2}+2wx+1)}}}} 2 [ ( 1 − v 2 ) ( 1 − w 2 ) − v w + 1 ] − 1 / 2 × {\displaystyle {\sqrt {2}},円{\bigl [}{\sqrt {(1-v^{2})(1-w^{2})}}-v,円w+1{\bigr ]}^{-1/2}\times } {\displaystyle {\sqrt {2}},円{\bigl [}{\sqrt {(1-v^{2})(1-w^{2})}}-v,円w+1{\bigr ]}^{-1/2}\times }

× F { arcsin ⁡ [ 1 − w 2 ( x + v ) + 1 − v 2 ( x + w ) 1 − w 2 x 2 + 2 v x + 1 + 1 − v 2 x 2 + 2 w x + 1 ] ; v − w ( 1 − v 2 ) ( 1 − w 2 ) − v w + 1 } {\displaystyle \times F{\biggl \{}\arcsin {\biggl [}{\tfrac {{\sqrt {1-w^{2}}}(x+v)+{\sqrt {1-v^{2}}}(x+w)}{{\sqrt {1-w^{2}}}{\sqrt {x^{2}+2vx+1}}+{\sqrt {1-v^{2}}}{\sqrt {x^{2}+2wx+1}}}}{\biggr ]};{\tfrac {v-w}{{\sqrt {(1-v^{2})(1-w^{2})}}-v,円w+1}}{\biggr \}}} {\displaystyle \times F{\biggl \{}\arcsin {\biggl [}{\tfrac {{\sqrt {1-w^{2}}}(x+v)+{\sqrt {1-v^{2}}}(x+w)}{{\sqrt {1-w^{2}}}{\sqrt {x^{2}+2vx+1}}+{\sqrt {1-v^{2}}}{\sqrt {x^{2}+2wx+1}}}}{\biggr ]};{\tfrac {v-w}{{\sqrt {(1-v^{2})(1-w^{2})}}-v,円w+1}}{\biggr \}}}

Vollständige Elliptische Integrale

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Funktion f ( x ) {\displaystyle f(x)} {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} {\displaystyle F(x)}
1 x ( 1 − x 2 ) [ E ( x ) − ( 1 − x 2 ) K ( x ) ] {\displaystyle {\frac {1}{x(1-x^{2})}}[E(x)-(1-x^{2})K(x)]} {\displaystyle {\frac {1}{x(1-x^{2})}}[E(x)-(1-x^{2})K(x)]} K ( x ) = ∫ 0 π / 2 1 1 − x 2 sin ⁡ ( y ) 2 d y = ∑ n = 0 ∞ CBC ⁡ ( n ) 2 16 n x 2 n {\displaystyle K(x)=\int _{0}^{\pi /2}{\frac {1}{\sqrt {1-x^{2}\sin(y)^{2}}}},円\mathrm {d} y=\sum _{n=0}^{\infty }{\frac {\operatorname {CBC} (n)^{2}}{16^{n}}}x^{2n}} {\displaystyle K(x)=\int _{0}^{\pi /2}{\frac {1}{\sqrt {1-x^{2}\sin(y)^{2}}}},円\mathrm {d} y=\sum _{n=0}^{\infty }{\frac {\operatorname {CBC} (n)^{2}}{16^{n}}}x^{2n}}
1 x [ E ( x ) − K ( x ) ] {\displaystyle {\frac {1}{x}}[E(x)-K(x)]} {\displaystyle {\frac {1}{x}}[E(x)-K(x)]} E ( x ) = ∫ 0 π / 2 1 − x 2 sin ⁡ ( y ) 2 d y = ∑ n = 0 ∞ CBC ⁡ ( n ) 2 16 n ( 1 − 2 n ) x 2 n {\displaystyle E(x)=\int _{0}^{\pi /2}{\sqrt {1-x^{2}\sin(y)^{2}}},円\mathrm {d} y=\sum _{n=0}^{\infty }{\frac {\operatorname {CBC} (n)^{2}}{16^{n}(1-2n)}}x^{2n}} {\displaystyle E(x)=\int _{0}^{\pi /2}{\sqrt {1-x^{2}\sin(y)^{2}}},円\mathrm {d} y=\sum _{n=0}^{\infty }{\frac {\operatorname {CBC} (n)^{2}}{16^{n}(1-2n)}}x^{2n}}
K ( x ) {\displaystyle K(x)} {\displaystyle K(x)} ∫ 0 1 arcsin ⁡ ( x y ) y 1 − y 2 d y {\displaystyle \int _{0}^{1}{\frac {\arcsin(xy)}{y{\sqrt {1-y^{2}}}}},円\mathrm {d} y} {\displaystyle \int _{0}^{1}{\frac {\arcsin(xy)}{y{\sqrt {1-y^{2}}}}},円\mathrm {d} y}
E ( x ) {\displaystyle E(x)} {\displaystyle E(x)} ∫ 0 1 [ arcsin ⁡ ( x y ) 2 y 1 − y 2 + x 1 − x 2 y 2 2 1 − y 2 ] d y {\displaystyle \int _{0}^{1}\left[{\frac {\arcsin(xy)}{2y{\sqrt {1-y^{2}}}}}+{\frac {x{\sqrt {1-x^{2}y^{2}}}}{2{\sqrt {1-y^{2}}}}}\right],円\mathrm {d} y} {\displaystyle \int _{0}^{1}\left[{\frac {\arcsin(xy)}{2y{\sqrt {1-y^{2}}}}}+{\frac {x{\sqrt {1-x^{2}y^{2}}}}{2{\sqrt {1-y^{2}}}}}\right],円\mathrm {d} y}
K ( x 2 ) {\displaystyle K(x^{2})} {\displaystyle K(x^{2})} ∫ 0 1 2 arcsl ⁡ ( x y ) 1 − y 4 d y {\displaystyle \int _{0}^{1}{\frac {2\operatorname {arcsl} (xy)}{\sqrt {1-y^{4}}}},円\mathrm {d} y} {\displaystyle \int _{0}^{1}{\frac {2\operatorname {arcsl} (xy)}{\sqrt {1-y^{4}}}},円\mathrm {d} y}
E ( x 2 ) {\displaystyle E(x^{2})} {\displaystyle E(x^{2})} ∫ 0 1 4 arcsl ⁡ ( x y ) 3 1 − y 4 + 2 x y 1 − x 4 y 4 3 1 − y 4 d y {\displaystyle \int _{0}^{1}{\frac {4\operatorname {arcsl} (xy)}{3{\sqrt {1-y^{4}}}}}+{\frac {2xy{\sqrt {1-x^{4}y^{4}}}}{3{\sqrt {1-y^{4}}}}},円\mathrm {d} y} {\displaystyle \int _{0}^{1}{\frac {4\operatorname {arcsl} (xy)}{3{\sqrt {1-y^{4}}}}}+{\frac {2xy{\sqrt {1-x^{4}y^{4}}}}{3{\sqrt {1-y^{4}}}}},円\mathrm {d} y}
π 2 q ( x ) 2 x ( 1 − x 2 ) K ( x ) 2 {\displaystyle {\frac {\pi ^{2}q(x)}{2x(1-x^{2})K(x)^{2}}}} {\displaystyle {\frac {\pi ^{2}q(x)}{2x(1-x^{2})K(x)^{2}}}} q ( x ) = exp ⁡ [ − π K ( 1 − x 2 ) / K ( x ) ] {\displaystyle q(x)=\exp {\bigl [}-\pi ,円K({\sqrt {1-x^{2}}})/K(x){\bigr ]}} {\displaystyle q(x)=\exp {\bigl [}-\pi ,円K({\sqrt {1-x^{2}}})/K(x){\bigr ]}}

Amplitudenfunktionen und lemniskatische Funktionen

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Funktion f ( x ) {\displaystyle f(x)} {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} {\displaystyle F(x)}
sl ⁡ ( x ) {\displaystyle \operatorname {sl} (x)} {\displaystyle \operatorname {sl} (x)} − arctan ⁡ [ cl ⁡ ( x ) ] {\displaystyle -\arctan[\operatorname {cl} (x)]} {\displaystyle -\arctan[\operatorname {cl} (x)]}
cl ⁡ ( x ) {\displaystyle \operatorname {cl} (x)} {\displaystyle \operatorname {cl} (x)} arctan ⁡ [ sl ⁡ ( x ) ] {\displaystyle \arctan[\operatorname {sl} (x)]} {\displaystyle \arctan[\operatorname {sl} (x)]}
sn ⁡ ( x ; k ) = sin ⁡ [ am ⁡ ( x ; k ) ] {\displaystyle \operatorname {sn} (x;k)=\sin {\bigl [}\operatorname {am} (x;k){\bigr ]}} {\displaystyle \operatorname {sn} (x;k)=\sin {\bigl [}\operatorname {am} (x;k){\bigr ]}} − 1 k artanh ⁡ [ k cd ⁡ ( x ; k ) ] {\displaystyle -{\frac {1}{k}}\operatorname {artanh} [k,円\operatorname {cd} (x;k)]} {\displaystyle -{\frac {1}{k}}\operatorname {artanh} [k,円\operatorname {cd} (x;k)]}
cn ⁡ ( x ; k ) = cos ⁡ [ am ⁡ ( x ; k ) ] {\displaystyle \operatorname {cn} (x;k)=\cos {\bigl [}\operatorname {am} (x;k){\bigr ]}} {\displaystyle \operatorname {cn} (x;k)=\cos {\bigl [}\operatorname {am} (x;k){\bigr ]}} 1 k arcsin ⁡ [ k sn ⁡ ( x ; k ) ] {\displaystyle {\frac {1}{k}}\operatorname {arcsin} [k,円\operatorname {sn} (x;k)]} {\displaystyle {\frac {1}{k}}\operatorname {arcsin} [k,円\operatorname {sn} (x;k)]}
dn ⁡ ( x ; k ) {\displaystyle \operatorname {dn} (x;k)} {\displaystyle \operatorname {dn} (x;k)} am ⁡ ( x ; k ) {\displaystyle \operatorname {am} (x;k)} {\displaystyle \operatorname {am} (x;k)}
cn ⁡ ( x ; k ) dn ⁡ ( x ; k ) {\displaystyle \operatorname {cn} (x;k)\operatorname {dn} (x;k)} {\displaystyle \operatorname {cn} (x;k)\operatorname {dn} (x;k)} sn ⁡ ( x ; k ) {\displaystyle \operatorname {sn} (x;k)} {\displaystyle \operatorname {sn} (x;k)}
− sn ⁡ ( x ; k ) dn ⁡ ( x ; k ) {\displaystyle -\operatorname {sn} (x;k)\operatorname {dn} (x;k)} {\displaystyle -\operatorname {sn} (x;k)\operatorname {dn} (x;k)} cn ⁡ ( x ; k ) {\displaystyle \operatorname {cn} (x;k)} {\displaystyle \operatorname {cn} (x;k)}
− k 2 sn ⁡ ( x ; k ) cn ⁡ ( x ; k ) {\displaystyle -k^{2}\operatorname {sn} (x;k)\operatorname {cn} (x;k)} {\displaystyle -k^{2}\operatorname {sn} (x;k)\operatorname {cn} (x;k)} dn ⁡ ( x ; k ) {\displaystyle \operatorname {dn} (x;k)} {\displaystyle \operatorname {dn} (x;k)}
zn ⁡ ( x ; k ) = E [ am ⁡ ( x ; k ) ; k ] − E ( k ) x K ( k ) {\displaystyle \operatorname {zn} (x;k)=E{\bigl [}\operatorname {am} (x;k);k{\bigr ]}-{\frac {E(k),円x}{K(k)}}} {\displaystyle \operatorname {zn} (x;k)=E{\bigl [}\operatorname {am} (x;k);k{\bigr ]}-{\frac {E(k),円x}{K(k)}}} ln ⁡ { ϑ 01 [ π 2 K ( k ) − 1 x ; q ( k ) ] } {\displaystyle \ln {\bigl \{}\vartheta _{01}{\bigl [}{\frac {\pi }{2}},円K(k)^{-1}x;q(k){\bigr ]}{\bigr \}}} {\displaystyle \ln {\bigl \{}\vartheta _{01}{\bigl [}{\frac {\pi }{2}},円K(k)^{-1}x;q(k){\bigr ]}{\bigr \}}}

Jacobische Thetafunktionen

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Funktion f ( x ) {\displaystyle f(x)} {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} {\displaystyle F(x)}
1 2 π x ϑ 10 ( x ) ϑ 00 ( x ) 2 E [ ϑ 10 ( x ) 2 ϑ 00 ( x ) 2 ] {\displaystyle {\frac {1}{2\pi x}}\vartheta _{10}(x)\vartheta _{00}(x)^{2}E\left[{\frac {\vartheta _{10}(x)^{2}}{\vartheta _{00}(x)^{2}}}\right]} {\displaystyle {\frac {1}{2\pi x}}\vartheta _{10}(x)\vartheta _{00}(x)^{2}E\left[{\frac {\vartheta _{10}(x)^{2}}{\vartheta _{00}(x)^{2}}}\right]} ϑ 10 ( x ) {\displaystyle \vartheta _{10}(x)} {\displaystyle \vartheta _{10}(x)}
ϑ 00 ( x ) [ ϑ 00 ( x ) 2 + ϑ 01 ( x ) 2 ] × {\displaystyle \vartheta _{00}(x)[\vartheta _{00}(x)^{2}+\vartheta _{01}(x)^{2}]\times } {\displaystyle \vartheta _{00}(x)[\vartheta _{00}(x)^{2}+\vartheta _{01}(x)^{2}]\times }

× { 1 2 π x E [ ϑ 00 ( x ) 2 − ϑ 01 ( x ) 2 ϑ 00 ( x ) 2 + ϑ 01 ( x ) 2 ] − ϑ 01 ( x ) 2 4 x } {\displaystyle \times \left\{{\frac {1}{2\pi x}}E\left[{\frac {\vartheta _{00}(x)^{2}-\vartheta _{01}(x)^{2}}{\vartheta _{00}(x)^{2}+\vartheta _{01}(x)^{2}}}\right]-{\frac {\vartheta _{01}(x)^{2}}{4x}}\right\}} {\displaystyle \times \left\{{\frac {1}{2\pi x}}E\left[{\frac {\vartheta _{00}(x)^{2}-\vartheta _{01}(x)^{2}}{\vartheta _{00}(x)^{2}+\vartheta _{01}(x)^{2}}}\right]-{\frac {\vartheta _{01}(x)^{2}}{4x}}\right\}}

ϑ 00 ( x ) {\displaystyle \vartheta _{00}(x)} {\displaystyle \vartheta _{00}(x)}
ϑ 01 ( x ) [ ϑ 00 ( x ) 2 + ϑ 01 ( x ) 2 ] × {\displaystyle \vartheta _{01}(x)[\vartheta _{00}(x)^{2}+\vartheta _{01}(x)^{2}]\times } {\displaystyle \vartheta _{01}(x)[\vartheta _{00}(x)^{2}+\vartheta _{01}(x)^{2}]\times }

× { 1 2 π x E [ ϑ 00 ( x ) 2 − ϑ 01 ( x ) 2 ϑ 00 ( x ) 2 + ϑ 01 ( x ) 2 ] − ϑ 00 ( x ) 2 4 x } {\displaystyle \times \left\{{\frac {1}{2\pi x}}E\left[{\frac {\vartheta _{00}(x)^{2}-\vartheta _{01}(x)^{2}}{\vartheta _{00}(x)^{2}+\vartheta _{01}(x)^{2}}}\right]-{\frac {\vartheta _{00}(x)^{2}}{4x}}\right\}} {\displaystyle \times \left\{{\frac {1}{2\pi x}}E\left[{\frac {\vartheta _{00}(x)^{2}-\vartheta _{01}(x)^{2}}{\vartheta _{00}(x)^{2}+\vartheta _{01}(x)^{2}}}\right]-{\frac {\vartheta _{00}(x)^{2}}{4x}}\right\}}

ϑ 01 ( x ) {\displaystyle \vartheta _{01}(x)} {\displaystyle \vartheta _{01}(x)}
ϑ 00 ( x ) {\displaystyle \vartheta _{00}(x)} {\displaystyle \vartheta _{00}(x)} x + 2 ∑ n = 1 ∞ x n 2 + 1 n 2 + 1 {\displaystyle x+2\sum _{n=1}^{\infty }{\frac {x^{n^{2}+1}}{n^{2}+1}}} {\displaystyle x+2\sum _{n=1}^{\infty }{\frac {x^{n^{2}+1}}{n^{2}+1}}}
ϑ 01 ( x ) {\displaystyle \vartheta _{01}(x)} {\displaystyle \vartheta _{01}(x)} x + 2 ∑ n = 1 ∞ ( − 1 ) n x n 2 + 1 n 2 + 1 {\displaystyle x+2\sum _{n=1}^{\infty }{\frac {(-1)^{n}x^{n^{2}+1}}{n^{2}+1}}} {\displaystyle x+2\sum _{n=1}^{\infty }{\frac {(-1)^{n}x^{n^{2}+1}}{n^{2}+1}}}
ϑ 10 ( x ) ϑ 01 ( x ) 4 4 x ϑ 00 ( x ) {\displaystyle {\frac {\vartheta _{10}(x)\vartheta _{01}(x)^{4}}{4x,円\vartheta _{00}(x)}}} {\displaystyle {\frac {\vartheta _{10}(x)\vartheta _{01}(x)^{4}}{4x,円\vartheta _{00}(x)}}} ϑ 10 ( x ) ϑ 00 ( x ) {\displaystyle {\frac {\vartheta _{10}(x)}{\vartheta _{00}(x)}}} {\displaystyle {\frac {\vartheta _{10}(x)}{\vartheta _{00}(x)}}}
ϑ 10 ( x ) ϑ 00 ( x ) 4 4 x ϑ 01 ( x ) {\displaystyle {\frac {\vartheta _{10}(x)\vartheta _{00}(x)^{4}}{4x,円\vartheta _{01}(x)}}} {\displaystyle {\frac {\vartheta _{10}(x)\vartheta _{00}(x)^{4}}{4x,円\vartheta _{01}(x)}}} ϑ 10 ( x ) ϑ 01 ( x ) {\displaystyle {\frac {\vartheta _{10}(x)}{\vartheta _{01}(x)}}} {\displaystyle {\frac {\vartheta _{10}(x)}{\vartheta _{01}(x)}}}
ϑ 00 ( x ) 5 − ϑ 00 ( x ) ϑ 01 ( x ) 4 4 x ϑ 01 ( x ) {\displaystyle {\frac {\vartheta _{00}(x)^{5}-\vartheta _{00}(x)\vartheta _{01}(x)^{4}}{4x,円\vartheta _{01}(x)}}} {\displaystyle {\frac {\vartheta _{00}(x)^{5}-\vartheta _{00}(x)\vartheta _{01}(x)^{4}}{4x,円\vartheta _{01}(x)}}} ϑ 00 ( x ) ϑ 01 ( x ) {\displaystyle {\frac {\vartheta _{00}(x)}{\vartheta _{01}(x)}}} {\displaystyle {\frac {\vartheta _{00}(x)}{\vartheta _{01}(x)}}}
ϑ 00 [ exp ⁡ ( − x ) ] {\displaystyle \vartheta _{00}{\bigl [}\exp(-x){\bigr ]}} {\displaystyle \vartheta _{00}{\bigl [}\exp(-x){\bigr ]}} x + ∑ n = 1 ∞ 2 n 2 [ 1 − exp ⁡ ( − n 2 x ) ] {\displaystyle x+\sum _{n=1}^{\infty }{\frac {2}{n^{2}}}{\bigl [}1-\exp(-n^{2}x){\bigr ]}} {\displaystyle x+\sum _{n=1}^{\infty }{\frac {2}{n^{2}}}{\bigl [}1-\exp(-n^{2}x){\bigr ]}}
ϑ 01 [ exp ⁡ ( − x ) ] {\displaystyle \vartheta _{01}{\bigl [}\exp(-x){\bigr ]}} {\displaystyle \vartheta _{01}{\bigl [}\exp(-x){\bigr ]}} x + ∑ n = 1 ∞ 2 n 2 ( − 1 ) n [ 1 − exp ⁡ ( − n 2 x ) ] {\displaystyle x+\sum _{n=1}^{\infty }{\frac {2}{n^{2}}}(-1)^{n}{\bigl [}1-\exp(-n^{2}x){\bigr ]}} {\displaystyle x+\sum _{n=1}^{\infty }{\frac {2}{n^{2}}}(-1)^{n}{\bigl [}1-\exp(-n^{2}x){\bigr ]}}
ϑ 01 [ exp ⁡ ( − x ) ] 2 {\displaystyle \vartheta _{01}{\bigl [}\exp(-x){\bigr ]}^{2}} {\displaystyle \vartheta _{01}{\bigl [}\exp(-x){\bigr ]}^{2}} x + ∑ n = 1 ∞ 2 n ( − 1 ) n gd ⁡ ( n x ) {\displaystyle x+\sum _{n=1}^{\infty }{\frac {2}{n}}(-1)^{n}\operatorname {gd} (n,円x)} {\displaystyle x+\sum _{n=1}^{\infty }{\frac {2}{n}}(-1)^{n}\operatorname {gd} (n,円x)}

Polylogarithmische Funktionen

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Die nicht elementaren Stammfunktionen von transzendenten Funktionen logarithmischer und arkusfunktionaler Art sowie die stammfunktionale Verkettung dieser Stammfunktionen werden als Polylogarithmen bezeichnet. Über den Rang der Polylogarithmen entscheiden die Indexzahlen in Fußnotenposition. Bei Indexzahl Zwei liegt der Dilogarithmus vor, welcher direkt als Ursprungsstammfunktion des elementar beschaffenen Monologarithmus hervorgeht. Die Linearkombinationen aus den Standard-Polylogarithmen werden Legendresche Chifunktionen genannt. Die Bestandteile der Stammfunktionskette von den Kreisbogenmaßfunktionen werden als Arkusfunktionsintegrale wie beispielsweise als Arkustangensintegrale und Arkussinusintegrale bezeichnet. Die imaginären Gegenstücke zu den Legendreschen Chifunktionen werden akkurat durch die Arkustangensintegrale der Standardform gebildet. Die Polylogarithmen aus Exponentialfunktionsausdrücken werden Debyesche Funktionen genannt und spielen bei der statistischen Thermodynamik die essentielle Hauptrolle unter den Funktionen.

Polylogarithmen der Standardform

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Funktion f ( x ) {\displaystyle f(x)} {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} {\displaystyle F(x)}
1 1 − x {\displaystyle {\frac {1}{1-x}}} {\displaystyle {\frac {1}{1-x}}} Li 1 ⁡ ( x ) = ln ⁡ ( 1 1 − x ) {\displaystyle \operatorname {Li} _{1}(x)=\ln {\bigl (}{\frac {1}{1-x}}{\bigr )}} {\displaystyle \operatorname {Li} _{1}(x)=\ln {\bigl (}{\frac {1}{1-x}}{\bigr )}}
1 x ln ⁡ ( 1 1 − x ) {\displaystyle {\frac {1}{x}}\ln {\bigl (}{\frac {1}{1-x}}{\bigr )}} {\displaystyle {\frac {1}{x}}\ln {\bigl (}{\frac {1}{1-x}}{\bigr )}} Li 2 ⁡ ( x ) {\displaystyle \operatorname {Li} _{2}(x)} {\displaystyle \operatorname {Li} _{2}(x)}
1 x Li 2 ⁡ ( x ) {\displaystyle {\frac {1}{x}}\operatorname {Li} _{2}(x)} {\displaystyle {\frac {1}{x}}\operatorname {Li} _{2}(x)} Li 3 ⁡ ( x ) {\displaystyle \operatorname {Li} _{3}(x)} {\displaystyle \operatorname {Li} _{3}(x)}
1 x Li n ⁡ ( x ) {\displaystyle {\frac {1}{x}}\operatorname {Li} _{n}(x)} {\displaystyle {\frac {1}{x}}\operatorname {Li} _{n}(x)} Li n + 1 ⁡ ( x ) {\displaystyle \operatorname {Li} _{n+1}(x)} {\displaystyle \operatorname {Li} _{n+1}(x)}
1 x ln ⁡ ( 1 1 − x ) 2 {\displaystyle {\frac {1}{x}}\ln {\bigl (}{\frac {1}{1-x}}{\bigr )}^{2}} {\displaystyle {\frac {1}{x}}\ln {\bigl (}{\frac {1}{1-x}}{\bigr )}^{2}} 2 Li 3 ⁡ ( x ) + 2 Li 3 ⁡ ( x x − 1 ) + 2 Li 1 ⁡ ( x ) Li 2 ⁡ ( x ) + 1 3 Li 1 ⁡ ( x ) 3 {\displaystyle 2\operatorname {Li} _{3}(x)+2\operatorname {Li} _{3}{\bigl (}{\frac {x}{x-1}}{\bigr )}+2\operatorname {Li} _{1}(x)\operatorname {Li} _{2}(x)+{\frac {1}{3}}\operatorname {Li} _{1}(x)^{3}} {\displaystyle 2\operatorname {Li} _{3}(x)+2\operatorname {Li} _{3}{\bigl (}{\frac {x}{x-1}}{\bigr )}+2\operatorname {Li} _{1}(x)\operatorname {Li} _{2}(x)+{\frac {1}{3}}\operatorname {Li} _{1}(x)^{3}}
1 x artanh ⁡ ( x ) {\displaystyle {\frac {1}{x}}\operatorname {artanh} (x)} {\displaystyle {\frac {1}{x}}\operatorname {artanh} (x)} χ 2 ( x ) = Li 2 ⁡ ( x ) − 1 4 Li 2 ⁡ ( x 2 ) = ∫ 0 1 arcsin ⁡ ( x y ) 1 − y 2 d y {\displaystyle \chi _{2}(x)=\operatorname {Li} _{2}(x)-{\frac {1}{4}}\operatorname {Li} _{2}(x^{2})=\int _{0}^{1}{\frac {\arcsin(xy)}{\sqrt {1-y^{2}}}},円\mathrm {d} y} {\displaystyle \chi _{2}(x)=\operatorname {Li} _{2}(x)-{\frac {1}{4}}\operatorname {Li} _{2}(x^{2})=\int _{0}^{1}{\frac {\arcsin(xy)}{\sqrt {1-y^{2}}}},円\mathrm {d} y}
1 x χ 2 ( x ) {\displaystyle {\frac {1}{x}}\chi _{2}(x)} {\displaystyle {\frac {1}{x}}\chi _{2}(x)} χ 3 ( x ) = Li 3 ⁡ ( x ) − 1 8 Li 3 ⁡ ( x 2 ) {\displaystyle \chi _{3}(x)=\operatorname {Li} _{3}(x)-{\frac {1}{8}}\operatorname {Li} _{3}(x^{2})} {\displaystyle \chi _{3}(x)=\operatorname {Li} _{3}(x)-{\frac {1}{8}}\operatorname {Li} _{3}(x^{2})}
ln ⁡ ( t x + u ) v x + w {\displaystyle {\frac {\ln(tx+u)}{vx+w}}} {\displaystyle {\frac {\ln(tx+u)}{vx+w}}} 1 v ln ⁡ ( u v − t w v ) ln ⁡ ( v x + w ) − 1 v Li 2 ⁡ ( − t v x + w u v − t w ) {\displaystyle {\frac {1}{v}}\ln {\biggl (}{\frac {uv-tw}{v}}{\biggr )}\ln(vx+w)-{\frac {1}{v}}\operatorname {Li} _{2}{\biggl (}-t,円{\frac {vx+w}{uv-tw}}{\biggr )}} {\displaystyle {\frac {1}{v}}\ln {\biggl (}{\frac {uv-tw}{v}}{\biggr )}\ln(vx+w)-{\frac {1}{v}}\operatorname {Li} _{2}{\biggl (}-t,円{\frac {vx+w}{uv-tw}}{\biggr )}} für den Fall t > 0 ∩ v > 0 ∩ t w − u v < 0 {\displaystyle t>0,円\cap ,円v>0,円\cap ,円tw-uv<0} {\displaystyle t>0,円\cap ,円v>0,円\cap ,円tw-uv<0}
ln ⁡ ( t x + u ) v x + w {\displaystyle {\frac {\ln(tx+u)}{vx+w}}} {\displaystyle {\frac {\ln(tx+u)}{vx+w}}} 1 v ln ⁡ ( t v x + w t w − u v ) ln ⁡ ( t x + u ) + 1 v Li 2 ⁡ ( − v t x + u t w − u v ) {\displaystyle {\frac {1}{v}}\ln {\biggl (}t,円{\frac {vx+w}{tw-uv}}{\biggr )}\ln(tx+u)+{\frac {1}{v}}\operatorname {Li} _{2}{\biggl (}-v,円{\frac {tx+u}{tw-uv}}{\biggr )}} {\displaystyle {\frac {1}{v}}\ln {\biggl (}t,円{\frac {vx+w}{tw-uv}}{\biggr )}\ln(tx+u)+{\frac {1}{v}}\operatorname {Li} _{2}{\biggl (}-v,円{\frac {tx+u}{tw-uv}}{\biggr )}} für den Fall t > 0 ∩ v > 0 ∩ t w − u v > 0 {\displaystyle t>0,円\cap ,円v>0,円\cap ,円tw-uv>0} {\displaystyle t>0,円\cap ,円v>0,円\cap ,円tw-uv>0}
artanh ⁡ ( x ) x 1 − x 2 {\displaystyle {\frac {\operatorname {artanh} (x)}{x{\sqrt {1-x^{2}}}}}} {\displaystyle {\frac {\operatorname {artanh} (x)}{x{\sqrt {1-x^{2}}}}}} 2 Li 2 ⁡ ( x 1 + 1 − x 2 ) − 1 2 Li 2 ⁡ ( 1 − 1 − x 2 1 + 1 − x 2 ) {\displaystyle 2\operatorname {Li} _{2}{\biggl (}{\frac {x}{1+{\sqrt {1-x^{2}}}}}{\biggr )}-{\frac {1}{2}}\operatorname {Li} _{2}{\biggl (}{\frac {1-{\sqrt {1-x^{2}}}}{1+{\sqrt {1-x^{2}}}}}{\biggr )}} {\displaystyle 2\operatorname {Li} _{2}{\biggl (}{\frac {x}{1+{\sqrt {1-x^{2}}}}}{\biggr )}-{\frac {1}{2}}\operatorname {Li} _{2}{\biggl (}{\frac {1-{\sqrt {1-x^{2}}}}{1+{\sqrt {1-x^{2}}}}}{\biggr )}}
artanh ⁡ ( x ) x ( 1 − x 2 ) {\displaystyle {\frac {\operatorname {artanh} (x)}{x(1-x^{2})}}} {\displaystyle {\frac {\operatorname {artanh} (x)}{x(1-x^{2})}}} 1 2 Li 2 ⁡ ( 2 x x + 1 ) + 1 2 artanh ⁡ ( x ) 2 {\displaystyle {\frac {1}{2}}\operatorname {Li} _{2}{\biggl (}{\frac {2x}{x+1}}{\biggr )}+{\frac {1}{2}}\operatorname {artanh} (x)^{2}} {\displaystyle {\frac {1}{2}}\operatorname {Li} _{2}{\biggl (}{\frac {2x}{x+1}}{\biggr )}+{\frac {1}{2}}\operatorname {artanh} (x)^{2}}
1 x arsinh ⁡ ( x ) {\displaystyle {\frac {1}{x}}\operatorname {arsinh} (x)} {\displaystyle {\frac {1}{x}}\operatorname {arsinh} (x)} 1 2 Li 2 ⁡ [ 1 − ( x 2 + 1 − x ) 2 ] + 1 2 arsinh ⁡ ( x ) 2 {\displaystyle {\frac {1}{2}}\operatorname {Li} _{2}[1-({\sqrt {x^{2}+1}}-x)^{2}]+{\frac {1}{2}}\operatorname {arsinh} (x)^{2}} {\displaystyle {\frac {1}{2}}\operatorname {Li} _{2}[1-({\sqrt {x^{2}+1}}-x)^{2}]+{\frac {1}{2}}\operatorname {arsinh} (x)^{2}}
1 x arsinh ⁡ ( x ) 2 {\displaystyle {\frac {1}{x}}\operatorname {arsinh} (x)^{2}} {\displaystyle {\frac {1}{x}}\operatorname {arsinh} (x)^{2}} 1 2 Li 3 ⁡ [ 1 − ( x 2 + 1 − x ) 2 ] + 1 2 Li 3 ⁡ [ 1 − ( x 2 + 1 + x ) 2 ] + arsinh ⁡ ( x ) Li 2 ⁡ [ 1 − ( x 2 + 1 − x ) 2 ] + arsinh ⁡ ( x ) 3 {\displaystyle {\frac {1}{2}}\operatorname {Li} _{3}[1-({\sqrt {x^{2}+1}}-x)^{2}]+{\frac {1}{2}}\operatorname {Li} _{3}[1-({\sqrt {x^{2}+1}}+x)^{2}]+\operatorname {arsinh} (x)\operatorname {Li} _{2}[1-({\sqrt {x^{2}+1}}-x)^{2}]+\operatorname {arsinh} (x)^{3}} {\displaystyle {\frac {1}{2}}\operatorname {Li} _{3}[1-({\sqrt {x^{2}+1}}-x)^{2}]+{\frac {1}{2}}\operatorname {Li} _{3}[1-({\sqrt {x^{2}+1}}+x)^{2}]+\operatorname {arsinh} (x)\operatorname {Li} _{2}[1-({\sqrt {x^{2}+1}}-x)^{2}]+\operatorname {arsinh} (x)^{3}}
arsinh ⁡ ( x ) x x 2 + 1 {\displaystyle {\frac {\operatorname {arsinh} (x)}{x{\sqrt {x^{2}+1}}}}} {\displaystyle {\frac {\operatorname {arsinh} (x)}{x{\sqrt {x^{2}+1}}}}} 2 Li 2 ⁡ ( x x 2 + 1 + 1 ) − 1 2 Li 2 ⁡ ( x 2 + 1 − 1 x 2 + 1 + 1 ) = ∫ 0 1 arctan ⁡ ( x 1 − y 2 ) 1 − y 2 d y {\displaystyle 2\operatorname {Li} _{2}{\biggl (}{\frac {x}{{\sqrt {x^{2}+1}}+1}}{\biggr )}-{\frac {1}{2}}\operatorname {Li} _{2}{\biggl (}{\frac {{\sqrt {x^{2}+1}}-1}{{\sqrt {x^{2}+1}}+1}}{\biggr )}=\int _{0}^{1}{\frac {\arctan(x{\sqrt {1-y^{2}}})}{\sqrt {1-y^{2}}}},円\mathrm {d} y} {\displaystyle 2\operatorname {Li} _{2}{\biggl (}{\frac {x}{{\sqrt {x^{2}+1}}+1}}{\biggr )}-{\frac {1}{2}}\operatorname {Li} _{2}{\biggl (}{\frac {{\sqrt {x^{2}+1}}-1}{{\sqrt {x^{2}+1}}+1}}{\biggr )}=\int _{0}^{1}{\frac {\arctan(x{\sqrt {1-y^{2}}})}{\sqrt {1-y^{2}}}},円\mathrm {d} y}
arsinh ⁡ ( x ) x ( x 2 + 1 ) {\displaystyle {\frac {\operatorname {arsinh} (x)}{x(x^{2}+1)}}} {\displaystyle {\frac {\operatorname {arsinh} (x)}{x(x^{2}+1)}}} Li 2 ⁡ ( x x 2 + 1 ) − 1 4 Li 2 ⁡ ( x 2 x 2 + 1 ) = Li 2 ⁡ [ 1 − ( x 2 + 1 − x ) 2 ] − 1 4 Li 2 ⁡ [ 1 − ( x 2 + 1 − x ) 4 ] {\displaystyle \operatorname {Li} _{2}{\biggl (}{\frac {x}{\sqrt {x^{2}+1}}}{\biggr )}-{\frac {1}{4}}\operatorname {Li} _{2}{\biggl (}{\frac {x^{2}}{x^{2}+1}}{\biggr )}=\operatorname {Li} _{2}[1-({\sqrt {x^{2}+1}}-x)^{2}]-{\frac {1}{4}}\operatorname {Li} _{2}[1-({\sqrt {x^{2}+1}}-x)^{4}]} {\displaystyle \operatorname {Li} _{2}{\biggl (}{\frac {x}{\sqrt {x^{2}+1}}}{\biggr )}-{\frac {1}{4}}\operatorname {Li} _{2}{\biggl (}{\frac {x^{2}}{x^{2}+1}}{\biggr )}=\operatorname {Li} _{2}[1-({\sqrt {x^{2}+1}}-x)^{2}]-{\frac {1}{4}}\operatorname {Li} _{2}[1-({\sqrt {x^{2}+1}}-x)^{4}]}

Arkustangensintegral und Arkussinusintegral

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Funktion f ( x ) {\displaystyle f(x)} {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} {\displaystyle F(x)}
1 x arctan ⁡ ( x ) {\displaystyle {\frac {1}{x}}\arctan(x)} {\displaystyle {\frac {1}{x}}\arctan(x)} Ti 2 ⁡ ( x ) {\displaystyle \operatorname {Ti} _{2}(x)} {\displaystyle \operatorname {Ti} _{2}(x)}
1 x Ti 2 ⁡ ( x ) {\displaystyle {\frac {1}{x}}\operatorname {Ti} _{2}(x)} {\displaystyle {\frac {1}{x}}\operatorname {Ti} _{2}(x)} Ti 3 ⁡ ( x ) {\displaystyle \operatorname {Ti} _{3}(x)} {\displaystyle \operatorname {Ti} _{3}(x)}
arctan ⁡ ( x ) x x 2 + 1 {\displaystyle {\frac {\arctan(x)}{x{\sqrt {x^{2}+1}}}}} {\displaystyle {\frac {\arctan(x)}{x{\sqrt {x^{2}+1}}}}} 2 Ti 2 ⁡ ( x x 2 + 1 + 1 ) {\displaystyle 2\operatorname {Ti} _{2}{\biggl (}{\frac {x}{{\sqrt {x^{2}+1}}+1}}{\biggr )}} {\displaystyle 2\operatorname {Ti} _{2}{\biggl (}{\frac {x}{{\sqrt {x^{2}+1}}+1}}{\biggr )}}
arctan ⁡ ( x ) x ( x 2 + 1 ) {\displaystyle {\frac {\arctan(x)}{x(x^{2}+1)}}} {\displaystyle {\frac {\arctan(x)}{x(x^{2}+1)}}} ∫ 0 1 arctan ⁡ ( x ) − 1 − y 2 arctan ⁡ ( x 1 − y 2 ) y 1 − y 2 d y {\displaystyle \int _{0}^{1}{\frac {\arctan(x)-{\sqrt {1-y^{2}}}\arctan(x{\sqrt {1-y^{2}}})}{y{\sqrt {1-y^{2}}}}},円\mathrm {d} y} {\displaystyle \int _{0}^{1}{\frac {\arctan(x)-{\sqrt {1-y^{2}}}\arctan(x{\sqrt {1-y^{2}}})}{y{\sqrt {1-y^{2}}}}},円\mathrm {d} y}

Debyesche Funktionen

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Funktion f ( x ) {\displaystyle f(x)} {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} {\displaystyle F(x)}
x n exp ⁡ ( x ) − 1 {\displaystyle {\frac {x^{n}}{\exp(x)-1}}} {\displaystyle {\frac {x^{n}}{\exp(x)-1}}} 1 n x n D n ( x ) {\displaystyle {\frac {1}{n}}x^{n},円\mathrm {D} _{n}(x)} {\displaystyle {\frac {1}{n}}x^{n},円\mathrm {D} _{n}(x)}
x exp ⁡ ( x ) − 1 {\displaystyle {\frac {x}{\exp(x)-1}}} {\displaystyle {\frac {x}{\exp(x)-1}}} x D 1 ( x ) = Li 2 ⁡ [ 1 − exp ⁡ ( − x ) ] {\displaystyle x,円\mathrm {D} _{1}(x)=\operatorname {Li} _{2}[1-\exp(-x)]} {\displaystyle x,円\mathrm {D} _{1}(x)=\operatorname {Li} _{2}[1-\exp(-x)]}
x 2 exp ⁡ ( x ) − 1 {\displaystyle {\frac {x^{2}}{\exp(x)-1}}} {\displaystyle {\frac {x^{2}}{\exp(x)-1}}} 1 2 x 2 D 2 ( x ) = 2 Li 3 ⁡ [ 1 − exp ⁡ ( − x ) ] + 2 Li 3 ⁡ [ 1 − exp ⁡ ( x ) ] + 2 x Li 2 ⁡ [ 1 − exp ⁡ ( − x ) ] + 1 3 x 3 {\displaystyle {\frac {1}{2}}x^{2},円\mathrm {D} _{2}(x)=2\operatorname {Li} _{3}[1-\exp(-x)]+2\operatorname {Li} _{3}[1-\exp(x)]+2x\operatorname {Li} _{2}[1-\exp(-x)]+{\frac {1}{3}}x^{3}} {\displaystyle {\frac {1}{2}}x^{2},円\mathrm {D} _{2}(x)=2\operatorname {Li} _{3}[1-\exp(-x)]+2\operatorname {Li} _{3}[1-\exp(x)]+2x\operatorname {Li} _{2}[1-\exp(-x)]+{\frac {1}{3}}x^{3}}
1 x arsinh ⁡ ( x ) n {\displaystyle {\frac {1}{x}}\operatorname {arsinh} (x)^{n}} {\displaystyle {\frac {1}{x}}\operatorname {arsinh} (x)^{n}} 1 n arsinh ⁡ ( x ) n D n [ 2 arsinh ⁡ ( x ) ] + 1 n + 1 arsinh ⁡ ( x ) n + 1 {\displaystyle {\frac {1}{n}}\operatorname {arsinh} (x)^{n},円\mathrm {D} _{n}{\bigl [}2\operatorname {arsinh} (x){\bigr ]}+{\frac {1}{n+1}}\operatorname {arsinh} (x)^{n+1}} {\displaystyle {\frac {1}{n}}\operatorname {arsinh} (x)^{n},円\mathrm {D} _{n}{\bigl [}2\operatorname {arsinh} (x){\bigr ]}+{\frac {1}{n+1}}\operatorname {arsinh} (x)^{n+1}}
1 x artanh ⁡ ( x ) n {\displaystyle {\frac {1}{x}}\operatorname {artanh} (x)^{n}} {\displaystyle {\frac {1}{x}}\operatorname {artanh} (x)^{n}} 1 n artanh ⁡ ( x ) n { 2 D n [ 2 artanh ⁡ ( x ) ] − D n [ 4 artanh ⁡ ( x ) ] } {\displaystyle {\frac {1}{n}}\operatorname {artanh} (x)^{n}{\bigl \{}2,円\mathrm {D} _{n}{\bigl [}2\operatorname {artanh} (x){\bigr ]}-\mathrm {D} _{n}{\bigl [}4\operatorname {artanh} (x){\bigr ]}{\bigr \}}} {\displaystyle {\frac {1}{n}}\operatorname {artanh} (x)^{n}{\bigl \{}2,円\mathrm {D} _{n}{\bigl [}2\operatorname {artanh} (x){\bigr ]}-\mathrm {D} _{n}{\bigl [}4\operatorname {artanh} (x){\bigr ]}{\bigr \}}}

Riemannsche und Dirichletsche Funktionen

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Funktion f ( x ) {\displaystyle f(x)} {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} {\displaystyle F(x)}
ζ ( x ) − x + 1 2 x − 2 {\displaystyle \zeta (x)-{\frac {x+1}{2x-2}}} {\displaystyle \zeta (x)-{\frac {x+1}{2x-2}}} ∫ 0 ∞ 4 ( y 2 + 1 ) x / 2 arctan ⁡ ( y ) − 4 arctan ⁡ ( y ) cos ⁡ [ x arctan ⁡ ( y ) ] − 2 ln ⁡ ( y 2 + 1 ) sin ⁡ [ x arctan ⁡ ( y ) ] ( y 2 + 1 ) x / 2 [ ln ⁡ ( y 2 + 1 ) 2 + 4 arctan ⁡ ( y ) 2 ] exp ⁡ ( π y ) sinh ⁡ ( π y ) d y {\displaystyle \int _{0}^{\infty }{\frac {4(y^{2}+1)^{x/2}\arctan(y)-4\arctan(y)\cos[x\arctan(y)]-2\ln(y^{2}+1)\sin[x\arctan(y)]}{(y^{2}+1)^{x/2}[\ln(y^{2}+1)^{2}+4\arctan(y)^{2}]\exp(\pi y)\sinh(\pi y)}},円\mathrm {d} y} {\displaystyle \int _{0}^{\infty }{\frac {4(y^{2}+1)^{x/2}\arctan(y)-4\arctan(y)\cos[x\arctan(y)]-2\ln(y^{2}+1)\sin[x\arctan(y)]}{(y^{2}+1)^{x/2}[\ln(y^{2}+1)^{2}+4\arctan(y)^{2}]\exp(\pi y)\sinh(\pi y)}},円\mathrm {d} y}
λ ( x ) − x 2 x − 2 {\displaystyle \lambda (x)-{\frac {x}{2x-2}}} {\displaystyle \lambda (x)-{\frac {x}{2x-2}}} ∫ 0 ∞ 2 ( y 2 + 1 ) x / 2 arctan ⁡ ( y ) − 2 arctan ⁡ ( y ) cos ⁡ [ x arctan ⁡ ( y ) ] − ln ⁡ ( y 2 + 1 ) sin ⁡ [ x arctan ⁡ ( y ) ] ( y 2 + 1 ) x / 2 [ ln ⁡ ( y 2 + 1 ) 2 + 4 arctan ⁡ ( y ) 2 ] exp ⁡ ( π y / 2 ) sinh ⁡ ( π y / 2 ) d y {\displaystyle \int _{0}^{\infty }{\frac {2(y^{2}+1)^{x/2}\arctan(y)-2\arctan(y)\cos[x\arctan(y)]-\ln(y^{2}+1)\sin[x\arctan(y)]}{(y^{2}+1)^{x/2}[\ln(y^{2}+1)^{2}+4\arctan(y)^{2}]\exp(\pi y/2)\sinh(\pi y/2)}},円\mathrm {d} y} {\displaystyle \int _{0}^{\infty }{\frac {2(y^{2}+1)^{x/2}\arctan(y)-2\arctan(y)\cos[x\arctan(y)]-\ln(y^{2}+1)\sin[x\arctan(y)]}{(y^{2}+1)^{x/2}[\ln(y^{2}+1)^{2}+4\arctan(y)^{2}]\exp(\pi y/2)\sinh(\pi y/2)}},円\mathrm {d} y}
η ( x ) − 1 2 {\displaystyle \eta (x)-{\frac {1}{2}}} {\displaystyle \eta (x)-{\frac {1}{2}}} ∫ 0 ∞ 4 ( y 2 + 1 ) x / 2 arctan ⁡ ( y ) − 4 arctan ⁡ ( y ) cos ⁡ [ x arctan ⁡ ( y ) ] − 2 ln ⁡ ( y 2 + 1 ) sin ⁡ [ x arctan ⁡ ( y ) ] ( y 2 + 1 ) x / 2 [ ln ⁡ ( y 2 + 1 ) 2 + 4 arctan ⁡ ( y ) 2 ] sinh ⁡ ( π y ) d y {\displaystyle \int _{0}^{\infty }{\frac {4(y^{2}+1)^{x/2}\arctan(y)-4\arctan(y)\cos[x\arctan(y)]-2\ln(y^{2}+1)\sin[x\arctan(y)]}{(y^{2}+1)^{x/2}[\ln(y^{2}+1)^{2}+4\arctan(y)^{2}]\sinh(\pi y)}},円\mathrm {d} y} {\displaystyle \int _{0}^{\infty }{\frac {4(y^{2}+1)^{x/2}\arctan(y)-4\arctan(y)\cos[x\arctan(y)]-2\ln(y^{2}+1)\sin[x\arctan(y)]}{(y^{2}+1)^{x/2}[\ln(y^{2}+1)^{2}+4\arctan(y)^{2}]\sinh(\pi y)}},円\mathrm {d} y}
β ( x ) − 1 2 {\displaystyle \beta (x)-{\frac {1}{2}}} {\displaystyle \beta (x)-{\frac {1}{2}}} ∫ 0 ∞ 2 ( y 2 + 1 ) x / 2 arctan ⁡ ( y ) − 2 arctan ⁡ ( y ) cos ⁡ [ x arctan ⁡ ( y ) ] − ln ⁡ ( y 2 + 1 ) sin ⁡ [ x arctan ⁡ ( y ) ] ( y 2 + 1 ) x / 2 [ ln ⁡ ( y 2 + 1 ) 2 + 4 arctan ⁡ ( y ) 2 ] sinh ⁡ ( π y / 2 ) d y {\displaystyle \int _{0}^{\infty }{\frac {2(y^{2}+1)^{x/2}\arctan(y)-2\arctan(y)\cos[x\arctan(y)]-\ln(y^{2}+1)\sin[x\arctan(y)]}{(y^{2}+1)^{x/2}[\ln(y^{2}+1)^{2}+4\arctan(y)^{2}]\sinh(\pi y/2)}},円\mathrm {d} y} {\displaystyle \int _{0}^{\infty }{\frac {2(y^{2}+1)^{x/2}\arctan(y)-2\arctan(y)\cos[x\arctan(y)]-\ln(y^{2}+1)\sin[x\arctan(y)]}{(y^{2}+1)^{x/2}[\ln(y^{2}+1)^{2}+4\arctan(y)^{2}]\sinh(\pi y/2)}},円\mathrm {d} y}

Verallgemeinerte Integrationsregeln

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Funktion f ( x ) {\displaystyle f(x)} {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} {\displaystyle F(x)}
u ′ ( x ) u ( x ) {\displaystyle {\frac {u'(x)}{u(x)}}} {\displaystyle {\frac {u'(x)}{u(x)}}} ln ⁡ | u ( x ) | {\displaystyle \ln \left|u(x)\right|} {\displaystyle \ln \left|u(x)\right|}
u ′ ( x ) ⋅ u ( x ) {\displaystyle u'(x)\cdot u(x)} {\displaystyle u'(x)\cdot u(x)} 1 2 ( u ( x ) ) 2 {\displaystyle {\tfrac {1}{2}}(u(x))^{2}} {\displaystyle {\tfrac {1}{2}}(u(x))^{2}}
u ′ ( x ) ⋅ ( u ( x ) ) n {\displaystyle u'(x)\cdot (u(x))^{n}} {\displaystyle u'(x)\cdot (u(x))^{n}} 1 n + 1 ( u ( x ) ) n + 1 {\displaystyle {\frac {1}{n+1}}(u(x))^{n+1}} {\displaystyle {\frac {1}{n+1}}(u(x))^{n+1}}

Lambertsche W-Funktion und invertierte Langevin-Funktion

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Funktion f ( x ) {\displaystyle f(x)} {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} {\displaystyle F(x)}
1 exp ⁡ [ W ( x ) ] + x {\displaystyle {\frac {1}{\exp[W(x)]+x}}} {\displaystyle {\frac {1}{\exp[W(x)]+x}}} W ( x ) {\displaystyle W(x)} {\displaystyle W(x)}
W ( x ) = 1 π ∫ − ∞ ∞ 1 y 2 + 1 ln ⁡ { 1 + x exp ⁡ [ y arccot ⁡ ( y ) ] y 2 + 1 arccot ⁡ ( y ) } d y {\displaystyle W(x)={\frac {1}{\pi }}\int _{-\infty }^{\infty }{\frac {1}{y^{2}+1}}\ln {\biggl \{}1+{\frac {x\exp {\bigl [}y\operatorname {arccot}(y){\bigr ]}}{{\sqrt {y^{2}+1}},円\operatorname {arccot}(y)}}{\biggr \}},円\mathrm {d} y} {\displaystyle W(x)={\frac {1}{\pi }}\int _{-\infty }^{\infty }{\frac {1}{y^{2}+1}}\ln {\biggl \{}1+{\frac {x\exp {\bigl [}y\operatorname {arccot}(y){\bigr ]}}{{\sqrt {y^{2}+1}},円\operatorname {arccot}(y)}}{\biggr \}},円\mathrm {d} y} exp ⁡ [ W ( x ) ] [ W ( x ) 2 − W ( x ) + 1 ] − 1 {\displaystyle \exp {\bigl [}W(x){\bigr ]}{\bigl [}W(x)^{2}-W(x)+1{\bigr ]}-1} {\displaystyle \exp {\bigl [}W(x){\bigr ]}{\bigl [}W(x)^{2}-W(x)+1{\bigr ]}-1}
1 W ( x ) + 1 = ∫ − ∞ ∞ 1 [ x exp ⁡ ( y ) − y ] 2 + π 2 d y {\displaystyle {\frac {1}{W(x)+1}}=\int _{-\infty }^{\infty }{\frac {1}{[x\exp(y)-y]^{2}+\pi ^{2}}}\mathrm {d} y} {\displaystyle {\frac {1}{W(x)+1}}=\int _{-\infty }^{\infty }{\frac {1}{[x\exp(y)-y]^{2}+\pi ^{2}}}\mathrm {d} y} exp ⁡ [ W ( x ) ] = x W ( x ) {\displaystyle \exp {\bigl [}W(x){\bigr ]}={\frac {x}{W(x)}}} {\displaystyle \exp {\bigl [}W(x){\bigr ]}={\frac {x}{W(x)}}}
L ⟨ − 1 ⟩ ( x ) 2 1 − L ⟨ − 1 ⟩ ( x ) 2 csch ⁡ [ L ⟨ − 1 ⟩ ( x ) ] 2 {\displaystyle {\frac {L^{\langle -1\rangle }(x)^{2}}{1-L^{\langle -1\rangle }(x)^{2}\operatorname {csch} {\bigl [}L^{\langle -1\rangle }(x){\bigr ]}^{2}}}} {\displaystyle {\frac {L^{\langle -1\rangle }(x)^{2}}{1-L^{\langle -1\rangle }(x)^{2}\operatorname {csch} {\bigl [}L^{\langle -1\rangle }(x){\bigr ]}^{2}}}} L ⟨ − 1 ⟩ ( x ) {\displaystyle L^{\langle -1\rangle }(x)} {\displaystyle L^{\langle -1\rangle }(x)}

Integralexponential- und Integrallogarithmusfunktion

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Die Integralexponentialfunktion und der Integrallogarithmus sind nicht elementar lösbar. Deswegen wird in den Stammfunktionen zusätzlich die Reihenentwicklung angegeben. Die als Integrationskonstante auftretende Konstante γ {\displaystyle \gamma } {\displaystyle \gamma } ist die Euler-Mascheroni-Konstante.

Funktion f ( x ) {\displaystyle f(x)} {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} {\displaystyle F(x)}
1 x e x {\displaystyle {\frac {1}{x}}e^{x}} {\displaystyle {\frac {1}{x}}e^{x}} Ei ⁡ ( x ) = γ + ln ⁡ | x | + ∑ k = 1 ∞ x k k ! ⋅ k {\displaystyle \operatorname {Ei} (x)=\gamma +\ln \left|x\right|+\sum _{k=1}^{\infty }{\frac {x^{k}}{k!\cdot k}}} {\displaystyle \operatorname {Ei} (x)=\gamma +\ln \left|x\right|+\sum _{k=1}^{\infty }{\frac {x^{k}}{k!\cdot k}}}
1 x n e x ( n ∈ N ) {\displaystyle {\frac {1}{x^{n}}}e^{x}\quad (n\in \mathbb {N} )} {\displaystyle {\frac {1}{x^{n}}}e^{x}\quad (n\in \mathbb {N} )} ( − ∑ k = 1 n − 1 ( k − 1 ) ! ( n − 1 ) ! x − k ) ⋅ e x + Ei ⁡ ( x ) = − ∑ k = 1 n − 1 x − ( n − k ) ( n − k ) ⋅ ( k − 1 ) ! + ln ⁡ | x | ( n − 1 ) ! + ∑ k = 0 ∞ x k + 1 ( k + 1 ) ⋅ ( n + k ) ! {\displaystyle \left(-\sum _{k=1}^{n-1}{\frac {(k-1)!}{(n-1)!}}x^{-k}\right)\cdot e^{x}+\operatorname {Ei} (x)=-\sum _{k=1}^{n-1}{\frac {x^{-(n-k)}}{(n-k)\cdot (k-1)!}}+{\frac {\ln |x|}{(n-1)!}}+\sum _{k=0}^{\infty }{\frac {x^{k+1}}{(k+1)\cdot (n+k)!}}} {\displaystyle \left(-\sum _{k=1}^{n-1}{\frac {(k-1)!}{(n-1)!}}x^{-k}\right)\cdot e^{x}+\operatorname {Ei} (x)=-\sum _{k=1}^{n-1}{\frac {x^{-(n-k)}}{(n-k)\cdot (k-1)!}}+{\frac {\ln |x|}{(n-1)!}}+\sum _{k=0}^{\infty }{\frac {x^{k+1}}{(k+1)\cdot (n+k)!}}}
log x ⁡ ( e ) = 1 ln ⁡ ( x ) {\displaystyle \log _{x}(e)={\frac {1}{\ln(x)}}} {\displaystyle \log _{x}(e)={\frac {1}{\ln(x)}}} li ⁡ ( x ) = Ei ⁡ ( ln ⁡ ( x ) ) = γ + ln ⁡ | ln ⁡ x | + ∑ k = 1 ∞ ( ln ⁡ x ) k k ⋅ k ! {\displaystyle \operatorname {li} (x)=\operatorname {Ei} (\ln(x))=\gamma +\ln \left|\ln x\right|+\sum _{k=1}^{\infty }{\frac {(\ln x)^{k}}{k\cdot k!}}} {\displaystyle \operatorname {li} (x)=\operatorname {Ei} (\ln(x))=\gamma +\ln \left|\ln x\right|+\sum _{k=1}^{\infty }{\frac {(\ln x)^{k}}{k\cdot k!}}}
log x ⁡ ( a ) = ln ⁡ ( a ) ln ⁡ ( x ) {\displaystyle \log _{x}(a)={\frac {\ln(a)}{\ln(x)}}} {\displaystyle \log _{x}(a)={\frac {\ln(a)}{\ln(x)}}} ln ⁡ ( a ) ⋅ li ⁡ ( x ) {\displaystyle \ln(a)\cdot \operatorname {li} (x)} {\displaystyle \ln(a)\cdot \operatorname {li} (x)}
ln ⁡ ( x ) ⋅ e x ( x > 0 ) {\displaystyle \ln(x)\cdot e^{x}\quad (x>0)} {\displaystyle \ln(x)\cdot e^{x}\quad (x>0)} ln ⁡ ( x ) ⋅ e x − Ei ⁡ ( x ) {\displaystyle \ln(x)\cdot e^{x}-\operatorname {Ei} (x)} {\displaystyle \ln(x)\cdot e^{x}-\operatorname {Ei} (x)}
ln ⁡ | ln ⁡ x | ( x > 0 , x ≠ 1 ) {\displaystyle \ln \left|\ln x\right|\quad (x>0,x\neq 1)} {\displaystyle \ln \left|\ln x\right|\quad (x>0,x\neq 1)} ln ⁡ | ln ⁡ x | − li ⁡ ( x ) {\displaystyle \ln \left|\ln x\right|-\operatorname {li} (x)} {\displaystyle \ln \left|\ln x\right|-\operatorname {li} (x)}

Integralkreisfunktionen und Gaußsches Fehlerintegral

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Funktion f ( x ) {\displaystyle f(x)} {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} {\displaystyle F(x)}
1 x sin ⁡ ( x ) {\displaystyle {\frac {1}{x}}\sin(x)} {\displaystyle {\frac {1}{x}}\sin(x)} Si ⁡ ( x ) = π 2 − cos ⁡ ( x ) ∫ 0 ∞ exp ⁡ ( − x y ) y 2 + 1 d y − sin ⁡ ( x ) ∫ 0 ∞ y exp ⁡ ( − x y ) y 2 + 1 d y {\displaystyle \operatorname {Si} (x)={\frac {\pi }{2}}-\cos(x)\int _{0}^{\infty }{\frac {\exp(-xy)}{y^{2}+1}},円\mathrm {d} y-\sin(x)\int _{0}^{\infty }{\frac {y\exp(-xy)}{y^{2}+1}},円\mathrm {d} y} {\displaystyle \operatorname {Si} (x)={\frac {\pi }{2}}-\cos(x)\int _{0}^{\infty }{\frac {\exp(-xy)}{y^{2}+1}},円\mathrm {d} y-\sin(x)\int _{0}^{\infty }{\frac {y\exp(-xy)}{y^{2}+1}},円\mathrm {d} y}
1 x cos ⁡ ( x ) {\displaystyle {\frac {1}{x}}\cos(x)} {\displaystyle {\frac {1}{x}}\cos(x)} Ci ⁡ ( x ) = sin ⁡ ( x ) ∫ 0 ∞ exp ⁡ ( − x y ) y 2 + 1 d y − cos ⁡ ( x ) ∫ 0 ∞ y exp ⁡ ( − x y ) y 2 + 1 d y {\displaystyle \operatorname {Ci} (x)=\sin(x)\int _{0}^{\infty }{\frac {\exp(-xy)}{y^{2}+1}},円\mathrm {d} y-\cos(x)\int _{0}^{\infty }{\frac {y\exp(-xy)}{y^{2}+1}},円\mathrm {d} y} {\displaystyle \operatorname {Ci} (x)=\sin(x)\int _{0}^{\infty }{\frac {\exp(-xy)}{y^{2}+1}},円\mathrm {d} y-\cos(x)\int _{0}^{\infty }{\frac {y\exp(-xy)}{y^{2}+1}},円\mathrm {d} y}
e − x 2 {\displaystyle \mathrm {e} ^{-x^{2}}} {\displaystyle \mathrm {e} ^{-x^{2}}} π 2 erf ⁡ ( x ) = 2 π erf ⁡ ( x ) ∫ 0 1 1 − exp ⁡ [ − x 2 ( y 2 + 1 ) ] y 2 + 1 d y {\displaystyle {\frac {\sqrt {\pi }}{2}}\operatorname {erf} (x)={\frac {2}{{\sqrt {\pi }}\operatorname {erf} (x)}}\int _{0}^{1}{\frac {1-\exp[-x^{2}(y^{2}+1)]}{y^{2}+1}},円\mathrm {d} y} {\displaystyle {\frac {\sqrt {\pi }}{2}}\operatorname {erf} (x)={\frac {2}{{\sqrt {\pi }}\operatorname {erf} (x)}}\int _{0}^{1}{\frac {1-\exp[-x^{2}(y^{2}+1)]}{y^{2}+1}},円\mathrm {d} y}[B 1]
e − a x 2 + b x + c {\displaystyle \mathrm {e} ^{-ax^{2}+bx+c}} {\displaystyle \mathrm {e} ^{-ax^{2}+bx+c}} π 2 a e b 2 4 a + c erf ⁡ ( a x − b 2 a ) {\displaystyle {\frac {\sqrt {\pi }}{2{\sqrt {a}}}}\mathrm {e} ^{{\frac {b^{2}}{4a}}+c}\operatorname {erf} \left({\sqrt {a}}x-{\frac {b}{2{\sqrt {a}}}}\right)} {\displaystyle {\frac {\sqrt {\pi }}{2{\sqrt {a}}}}\mathrm {e} ^{{\frac {b^{2}}{4a}}+c}\operatorname {erf} \left({\sqrt {a}}x-{\frac {b}{2{\sqrt {a}}}}\right)}[B 1]

Gammafunktion und Polygammafunktionen

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Funktion f ( x ) {\displaystyle f(x)} {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} {\displaystyle F(x)}
Π ( x ) = Γ ( x + 1 ) = x ! {\displaystyle \Pi (x)=\Gamma (x+1)=x!} {\displaystyle \Pi (x)=\Gamma (x+1)=x!} ∫ 0 ∞ y x − 1 ln ⁡ ( y ) exp ⁡ ( y ) d y {\displaystyle \int _{0}^{\infty }{\frac {y^{x}-1}{\ln(y)\exp(y)}},円\mathrm {d} y} {\displaystyle \int _{0}^{\infty }{\frac {y^{x}-1}{\ln(y)\exp(y)}},円\mathrm {d} y}
ln ⁡ [ Π ( x ) ] = ln ⁡ [ Γ ( x + 1 ) ] {\displaystyle \ln {\bigl [}\Pi (x){\bigr ]}=\ln {\bigl [}\Gamma (x+1){\bigr ]}} {\displaystyle \ln {\bigl [}\Pi (x){\bigr ]}=\ln {\bigl [}\Gamma (x+1){\bigr ]}} ln ⁡ [ hf ⁡ ( x ) ] − x 2 [ x + 1 − ln ⁡ ( 2 π ) ] = ( x + 1 ) ln ⁡ [ Π ( x ) ] − ln ⁡ [ sf ⁡ ( x ) ] − x 2 [ x + 1 − ln ⁡ ( 2 π ) ] {\displaystyle \ln {\bigl [}\operatorname {hf} (x){\bigr ]}-{\frac {x}{2}}{\bigl [}x+1-\ln(2,円\pi ){\bigr ]}=(x+1)\ln {\bigl [}\Pi (x){\bigr ]}-\ln {\bigl [}\operatorname {sf} (x){\bigr ]}-{\frac {x}{2}}{\bigl [}x+1-\ln(2,円\pi ){\bigr ]}} {\displaystyle \ln {\bigl [}\operatorname {hf} (x){\bigr ]}-{\frac {x}{2}}{\bigl [}x+1-\ln(2,円\pi ){\bigr ]}=(x+1)\ln {\bigl [}\Pi (x){\bigr ]}-\ln {\bigl [}\operatorname {sf} (x){\bigr ]}-{\frac {x}{2}}{\bigl [}x+1-\ln(2,円\pi ){\bigr ]}}
H ( x ) = γ + ψ ( x + 1 ) {\displaystyle \mathrm {H} (x)=\gamma +\psi (x+1)} {\displaystyle \mathrm {H} (x)=\gamma +\psi (x+1)}[B 2] γ x + ln ⁡ [ Π ( x ) ] = ∑ n = 1 ∞ [ x n − ln ⁡ ( 1 + x n ) ] = ∫ 0 ∞ exp ⁡ ( − x y ) + x y − 1 y [ exp ⁡ ( y ) − 1 ] d y {\displaystyle \gamma x+\ln {\bigl [}\Pi (x){\bigr ]}=\sum _{n=1}^{\infty }{\biggl [}{\frac {x}{n}}-\ln {\biggl (}1+{\frac {x}{n}}{\biggr )}{\biggr ]}=\int _{0}^{\infty }{\frac {\exp(-xy)+xy-1}{y{\bigl [}\exp(y)-1{\bigr ]}}},円\mathrm {d} y} {\displaystyle \gamma x+\ln {\bigl [}\Pi (x){\bigr ]}=\sum _{n=1}^{\infty }{\biggl [}{\frac {x}{n}}-\ln {\biggl (}1+{\frac {x}{n}}{\biggr )}{\biggr ]}=\int _{0}^{\infty }{\frac {\exp(-xy)+xy-1}{y{\bigl [}\exp(y)-1{\bigr ]}}},円\mathrm {d} y}
ψ 1 ( x + 1 ) = ∑ n = 1 ∞ 1 ( x + n ) 2 {\displaystyle \psi _{1}(x+1)=\sum _{n=1}^{\infty }{\frac {1}{(x+n)^{2}}}} {\displaystyle \psi _{1}(x+1)=\sum _{n=1}^{\infty }{\frac {1}{(x+n)^{2}}}} H ( x ) {\displaystyle \mathrm {H} (x)} {\displaystyle \mathrm {H} (x)}

Besselsche Funktionen und Airysche Funktionen

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Funktion f ( x ) {\displaystyle f(x)} {\displaystyle f(x)} Stammfunktion F ( x ) {\displaystyle F(x)} {\displaystyle F(x)}
I 0 ( x ) = ∑ n = 0 ∞ x 2 n 4 n ( n ! ) 2 {\displaystyle \mathrm {I} _{0}(x)=\sum _{n=0}^{\infty }{\frac {x^{2n}}{4^{n}(n!)^{2}}}} {\displaystyle \mathrm {I} _{0}(x)=\sum _{n=0}^{\infty }{\frac {x^{2n}}{4^{n}(n!)^{2}}}} ∫ 0 π 1 π csc ⁡ ( y ) sinh ⁡ [ x sin ⁡ ( y ) ] d y {\displaystyle \int _{0}^{\pi }{\frac {1}{\pi }}\csc(y)\sinh {\bigl [}x\sin(y){\bigr ]},円\mathrm {d} y} {\displaystyle \int _{0}^{\pi }{\frac {1}{\pi }}\csc(y)\sinh {\bigl [}x\sin(y){\bigr ]},円\mathrm {d} y}
J 0 ( x ) = ∑ n = 0 ∞ ( − 1 ) n x 2 n 4 n ( n ! ) 2 {\displaystyle \mathrm {J} _{0}(x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{4^{n}(n!)^{2}}}} {\displaystyle \mathrm {J} _{0}(x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{4^{n}(n!)^{2}}}} ∫ 0 π 1 π csc ⁡ ( y ) sin ⁡ [ x sin ⁡ ( y ) ] d y {\displaystyle \int _{0}^{\pi }{\frac {1}{\pi }}\csc(y)\sin {\bigl [}x\sin(y){\bigr ]},円\mathrm {d} y} {\displaystyle \int _{0}^{\pi }{\frac {1}{\pi }}\csc(y)\sin {\bigl [}x\sin(y){\bigr ]},円\mathrm {d} y}
A i ( x ) {\displaystyle \mathrm {Ai} (x)} {\displaystyle \mathrm {Ai} (x)} ∫ 0 ∞ 1 π y [ sin ⁡ ( 1 3 y 3 + x y ) − sin ⁡ ( 1 3 y 3 ) ] d y {\displaystyle \int _{0}^{\infty }{\frac {1}{\pi ,円y}}{\bigl [}\sin {\bigl (}{\frac {1}{3}}y^{3}+xy{\bigr )}-\sin {\bigl (}{\frac {1}{3}}y^{3}{\bigr )}{\bigr ]},円\mathrm {d} y} {\displaystyle \int _{0}^{\infty }{\frac {1}{\pi ,円y}}{\bigl [}\sin {\bigl (}{\frac {1}{3}}y^{3}+xy{\bigr )}-\sin {\bigl (}{\frac {1}{3}}y^{3}{\bigr )}{\bigr ]},円\mathrm {d} y}
B i ( x ) {\displaystyle \mathrm {Bi} (x)} {\displaystyle \mathrm {Bi} (x)} ∫ 0 ∞ 1 π y [ exp ⁡ ( − 1 3 y 3 + x y ) − exp ⁡ ( − 1 3 y 3 ) − cos ⁡ ( 1 3 y 3 + x y ) + cos ⁡ ( 1 3 y 3 ) ] d y {\displaystyle \int _{0}^{\infty }{\frac {1}{\pi ,円y}}{\bigl [}\exp {\bigl (}-{\frac {1}{3}}y^{3}+xy{\bigr )}-\exp {\bigl (}-{\frac {1}{3}}y^{3}{\bigr )}-\cos {\bigl (}{\frac {1}{3}}y^{3}+xy{\bigr )}+\cos {\bigl (}{\frac {1}{3}}y^{3}{\bigr )}{\bigr ]},円\mathrm {d} y} {\displaystyle \int _{0}^{\infty }{\frac {1}{\pi ,円y}}{\bigl [}\exp {\bigl (}-{\frac {1}{3}}y^{3}+xy{\bigr )}-\exp {\bigl (}-{\frac {1}{3}}y^{3}{\bigr )}-\cos {\bigl (}{\frac {1}{3}}y^{3}+xy{\bigr )}+\cos {\bigl (}{\frac {1}{3}}y^{3}{\bigr )}{\bigr ]},円\mathrm {d} y}
  1. ↑ a b erf {\displaystyle \operatorname {erf} } {\displaystyle \operatorname {erf} } ist die Fehlerfunktion
  2. ↑ H {\displaystyle \mathrm {H} } {\displaystyle \mathrm {H} } ist die Harmonische Reihe

Rekursionsformeln für weitere Stammfunktionen

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  • ∫ 1 ( x 2 + 1 ) n d x = 1 2 n − 2 ⋅ x ( x 2 + 1 ) n − 1 + 2 n − 3 2 n − 2 ⋅ ∫ 1 ( x 2 + 1 ) n − 1 d x , n ≥ 2 {\displaystyle \int {\frac {1}{(x^{2}+1)^{n}}},円\mathrm {d} x={\frac {1}{2n-2}}\cdot {\frac {x}{(x^{2}+1)^{n-1}}}+{\frac {2n-3}{2n-2}}\cdot \int {\frac {1}{(x^{2}+1)^{n-1}}},円\mathrm {d} x,\quad n\geq 2} {\displaystyle \int {\frac {1}{(x^{2}+1)^{n}}},円\mathrm {d} x={\frac {1}{2n-2}}\cdot {\frac {x}{(x^{2}+1)^{n-1}}}+{\frac {2n-3}{2n-2}}\cdot \int {\frac {1}{(x^{2}+1)^{n-1}}},円\mathrm {d} x,\quad n\geq 2}
  • ∫ sin n ⁡ ( x ) d x = n − 1 n ⋅ ∫ sin n − 2 ⁡ ( x ) d x − 1 n ⋅ cos ⁡ ( x ) ⋅ sin n − 1 ⁡ ( x ) , n ≥ 2 {\displaystyle \int \sin ^{n}(x),円\mathrm {d} x={\frac {n-1}{n}}\cdot \int \sin ^{n-2}(x),円\mathrm {d} x-{\frac {1}{n}}\cdot \cos(x)\cdot \sin ^{n-1}(x),\quad n\geq 2} {\displaystyle \int \sin ^{n}(x),円\mathrm {d} x={\frac {n-1}{n}}\cdot \int \sin ^{n-2}(x),円\mathrm {d} x-{\frac {1}{n}}\cdot \cos(x)\cdot \sin ^{n-1}(x),\quad n\geq 2}
  • ∫ cos n ⁡ ( x ) d x = n − 1 n ⋅ ∫ cos n − 2 ⁡ ( x ) d x + 1 n ⋅ sin ⁡ ( x ) ⋅ cos n − 1 ⁡ ( x ) , n ≥ 2 {\displaystyle \int \cos ^{n}(x),円\mathrm {d} x={\frac {n-1}{n}}\cdot \int \cos ^{n-2}(x),円\mathrm {d} x+{\frac {1}{n}}\cdot \sin(x)\cdot \cos ^{n-1}(x),\quad n\geq 2} {\displaystyle \int \cos ^{n}(x),円\mathrm {d} x={\frac {n-1}{n}}\cdot \int \cos ^{n-2}(x),円\mathrm {d} x+{\frac {1}{n}}\cdot \sin(x)\cdot \cos ^{n-1}(x),\quad n\geq 2}

Multiplikation von Stammfunktionen

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Für die Multiplikation zweier Stammfunktionen kann der Satz von Fubini in Kombination mit der Produktregel angewendet werden:

[ ∫ 0 w f ( x ) d x ] [ ∫ 0 w g ( x ) d x ] = ∫ 0 1 ∫ 0 w x f ( x ) g ( x y ) + x g ( x ) f ( x y ) d x d y {\displaystyle {\biggl [}\int _{0}^{w}f(x),円\mathrm {d} x{\biggr ]}{\biggl [}\int _{0}^{w}g(x),円\mathrm {d} x{\biggr ]}=\int _{0}^{1}\int _{0}^{w}xf(x)g(xy)+xg(x)f(xy),円\mathrm {d} x,円\mathrm {d} y} {\displaystyle {\biggl [}\int _{0}^{w}f(x),円\mathrm {d} x{\biggr ]}{\biggl [}\int _{0}^{w}g(x),円\mathrm {d} x{\biggr ]}=\int _{0}^{1}\int _{0}^{w}xf(x)g(xy)+xg(x)f(xy),円\mathrm {d} x,円\mathrm {d} y}
Abgerufen von „https://de.wikipedia.org/w/index.php?title=Tabelle_von_Ableitungs-_und_Stammfunktionen&oldid=251825405"