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Bessel functions, are canonical solutions of Bessel’s differential equation:
x^2 \\frac{d^2 y}{dx^2} + x \\frac{dy}{dx} + (x^2 - \\alpha^2) y = 0
(削除ここまで)for an arbitrary real or complex number (削除) (追記) (追記ここまで) (the order of the Bessel function). The most common and important special case is where \\alpha (削除ここまで)(削除) (追記) (追記ここまで) is an integer (in which case we call it ).\\alpha (削除ここまで)
J_\\alpha (削除ここまで)Bessel functions of the first kind, denoted (削除) (追記) (追記ここまで), are solutions of Bessel’s differential equation that are finite at the origin () for non-negative integer J_\\alpha(x) (削除ここまで)(削除) (追記) (追記ここまで), and diverge as \\alpha (削除ここまで)(削除) (追記) (追記ここまで) for negative non-integer x \\to 0 (削除ここまで)(削除) (追記) (追記ここまで). It is possible to define the function by its Taylor series expansion around :\\alpha (削除ここまで)
J_\\alpha(x) = \\sum_{m=0}^\\infty \\frac{(-1)^m}{m! \\Gamma(m+\\alpha+1)} {\\left({\\frac{x}{2}}\\right)}^{2m+\\alpha}
(削除ここまで)where (削除) (追記) (追記ここまで) is the gamma function, a generalization of the factorial function to non-integer values.\\Gamma(z) (削除ここまで)
For evaluating Bessel functions of the first kind in Fortran, see bessel_j0, bessel_j1, and bessel_jn.
Y_\\alpha (削除ここまで)Bessel functions of the second kind, denoted by (削除) (追記) (追記ここまで), are solutions of the Bessel differential equation. They are singular (infinite) at the origin ().Y_\\alpha(x) (削除ここまで)
For non-integer (削除) (追記) (追記ここまで), it is related to \\alpha (削除ここまで)(削除) (追記) (追記ここまで) by:J_\\alpha(x) (削除ここまで)
Y_\\alpha(x) = \\frac{J_\\alpha(x) \\cos(\\alpha\\pi) - J_{-\\alpha}(x)}{\\sin(\\alpha\\pi)}.
(削除ここまで)In the case of integer order , the function is defined by taking the limit as a non-integer (削除) (追記) (追記ここまで) tends to :\\alpha (削除ここまで)
Y_n(x) = \\lim_{\\alpha \\to n} Y_\\alpha(x),
(削除ここまで)which has the result (in integral form)
Y_n(x) =
\\frac{1}{\\pi} \\int_{0}^{\\pi} \\sin(x \\sin\\theta - n\\theta)d\\theta
- \\frac{1}{\\pi} \\int_{0}^{\\infty}
\\left[ e^{n t} + (-1)^n e^{-n t} \\right]
e^{-x \\sinh t} dt.
(削除ここまで)For evaluating Bessel functions of the second kind in Fortran, see bessel_y0, bessel_y1, and bessel_yn.