Loterre: Mathématiques: modified Bessel function

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modified Bessel function

The Bessel functions are valid even for complex arguments $x$ , and an important special case is that of a purely imaginary argument. In this case, the solutions to the Bessel equation are called the modified Bessel functions (or occasionally the hyperbolic Bessel functions) of the first and second kind and are defined as
${\\displaystyle {\egin{aligned}I_{\\alpha }(x)&=i^{-\\alpha }J_{\\alpha }(ix)=\\sum _{m=0}^{\\infty }{\ rac {1}{m!\\,\\Gamma (m+\\alpha +1)}}\\left({\ rac {x}{2}}\ ight)^{2m+\\alpha },\\\\K_{\\alpha }(x)&={\ rac {\\pi }{2}}{\ rac {I_{-\\alpha }(x)-I_{\\alpha }(x)}{\\sin \\alpha \\pi }},\\end{aligned}}}$
when $α$ is not an integer; when $α$ is an integer, then the limit is used. These are chosen to be real-valued for real and positive arguments $x$ . The series expansion for $I α (x)$ is thus similar to that for $J α (x)$ , but without the alternating $(-1) m$ factor.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Bessel_function#Modified_Bessel_functions)

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