: Mathematics: Bessel function

mathematical physics > special function > hypergeometric function > Bessel function

mathematical analysis > functional analysis > special function > hypergeometric function > Bessel function

Bessel function

essel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions $y (x)$ of Bessel's differential equation
${\\displaystyle x^{2}{\ rac {d^{2}y}{dx^{2}}}+x{\ rac {dy}{dx}}+\\left(x^{2}-\\alpha ^{2}\ ight)y=0}$
for an arbitrary complex number ${\\displaystyle \\alpha }$ , which represents the order of the Bessel function. Although ${\\displaystyle \\alpha }$ and ${\\displaystyle -\\alpha }$ produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of ${\\displaystyle \\alpha }$ .
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Bessel_function)

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