Bessel Function of the First Kind/Instances/Order 0

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Specific Instance of Bessel Functions of the First Kind

Let $\map {J_n} x$ denote the Bessel function of the first kind of order $n$.


\(\ds \map {J_0} x\) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {\paren {k!}^2} \paren {\dfrac x 2}^{2 k}\)
\(\ds \) \(=\) \(\ds 1 - \dfrac {x^2} {2^2} + \dfrac {x^4} {2^2 \times 4^2} - \dfrac {x^6} {2^2 \times 4^2 \times 6^2} + \dotsb\)


Proof

From Series Expansion of Bessel Function of the First Kind:

\(\ds \map {J_n} x\) \(=\) \(\ds \dfrac {x^n} {2^n \, \map \Gamma {n + 1} } \paren {1 - \dfrac {x^2} {2 \paren {2 n + 2} } + \dfrac {x^4} {2 \times 4 \paren {2 n + 2} \paren {2 n + 4} } - \cdots}\)
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {k! \, \map \Gamma {n + k + 1} } \paren {\dfrac x 2}^{n + 2 k}\)

where $n$ is not a (strictly) negative integer.


$0$ fits that category, and so:

\(\ds \map {J_0} x\) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {k! \, \map \Gamma {0 + k + 1} } \paren {\dfrac x 2}^{0 + 2 k}\)
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {k! \, \map \Gamma {k + 1} } \paren {\dfrac x 2}^{2 k}\)
\(\ds \) \(=\) \(\ds \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {\paren {k!}^2} \paren {\dfrac x 2}^{2 k}\) Gamma Function Extends Factorial


Or working directly upon the terms themselves:

\(\ds \map {J_0} x\) \(=\) \(\ds \dfrac {x^0} {2^0 \, \map \Gamma {0 + 1} } \paren {1 - \dfrac {x^2} {2 \paren {2 \times 0 + 2} } + \dfrac {x^4} {2 \times 4 \paren {2 \times 0 + 2} \paren {2 \times 0 + 4} } - \cdots}\)
\(\ds \) \(=\) \(\ds \dfrac 1 {1 \times \map \Gamma 1} \paren {1 - \dfrac {x^2} {2 \times 2} + \dfrac {x^4} {2 \times 4 \times 2 \times 4} - \cdots}\)
\(\ds \) \(=\) \(\ds 1 - \dfrac {x^2} {2^2} + \dfrac {x^4} {2^2 \times 4^2} - \cdots\)

$\blacksquare$


Sources

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