Hyperbolic Lemniscate Function
By analogy with the lemniscate functions, hyperbolic lemniscate functions can also be defined
arcsinhlemnx = [画像:int_0^x(1+t^4)^(1/2)dt]
(1)
= x_2F_1(-1/2,1/4;5/4;-x^4)
(2)
arccoshlemnx = [画像:int_x^1(1+t^4)^(1/2)dt]
(3)
= _2F_1(-1/2,1/4;5/4;-1)-x_2F_1(-1/2,1/4;5/4;-x^4).
(4)
where _2F_1(a,b;c;z) is a hypergeometric function.
Let 0<=theta<=pi/2 and 0<=v<=1, and write
(thetamu)/2 = [画像:int_0^v(dt)/(sqrt(1+t^4))]
(5)
= v_2F_1(1/4,1/2;5/4;-v^4),
(6)
where mu is the constant obtained by setting theta=pi/2 and v=1, which is given by
mu = [画像:2/piK(1/(sqrt(2)))]
(7)
= [画像:(sqrt(pi))/(Gamma^2(3/4)),]
(8)
with K(k) is a complete elliptic integral of the first kind. Ramanujan showed that
| [画像: 1/8pi-1/2tan^(-1)(v^2)=sum_(n=0)^infty((-1)^ncos[(2n+1)theta])/((2n+1)cosh[1/2(2n+1)pi]) ] |
(10)
|
![画像: 1/8pi-1/2tan^(-1)(v^2)=sum_(n=0)^infty((-1)^ncos[(2n+1)theta])/((2n+1)cosh[1/2(2n+1)pi])](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/HyperbolicLemniscateFunction/NumberedEquation2.svg){kind=link}
and
| ln((1+v)/(1-v))=ln[tan(1/4pi+1/2theta)]+4sum_(n=0)^infty((-1)^nsin[(2n+1)theta])/((2n+1)[e^((2n+1)pi)-1]) |
(11)
|
(Berndt 1994).
See also
Lemniscate FunctionExplore with Wolfram|Alpha
WolframAlpha
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References
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 255-258, 1994.Referenced on Wolfram|Alpha
Hyperbolic Lemniscate FunctionCite this as:
Weisstein, Eric W. "Hyperbolic Lemniscate Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HyperbolicLemniscateFunction.html