10 Bessel Functions Bessel and Hankel Functions 10.11 Analytic Continuation 10.13 Other Differential Equations

§10.12 Generating Function and Associated Series

Notes:: For (10.12.1) see Olver (1997b, pp. 55–56). For (10.12.2)–(10.12.6) set $t=e^{{i\theta}}$ and $ie^{{i\theta}}$ , and apply other straighforward substitutions, including differentiations with respect to $\theta$ in the case of (10.12.6). See also Watson (1944, pp. 22–23).
Keywords:: Bessel functions
Permalink:: http://dlmf.nist.gov/10.12

For $z\in\Complex$ and $t\in\Complex\setminus\{0\}$ ,

10.12.1 $e^{{\frac{1}{2}z(t-t^{{-1}})}}=\sum_{{m=-\infty}}^{\infty}t^{m}\mathop{J_{{m}}% \/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, : base of exponential function, : integer and : complex variable
A&S Ref:: 9.1.41
Referenced by:: §10.12, ¶ ‣ §10.23(iii), §10.35
Permalink:: http://dlmf.nist.gov/10.12.E1
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For $z,\theta\in\Complex$ ,

10.12.2

$\mathop{\cos\/}\nolimits\!\left(z\mathop{\sin\/}\nolimits\theta\right)=\mathop% {J_{{0}}\/}\nolimits\!\left(z\right)+2\sum_{{k=1}}^{\infty}\mathop{J_{{2k}}\/}% \nolimits\!\left(z\right)\mathop{\cos\/}\nolimits\!\left(2k\theta\right),$

$\mathop{\sin\/}\nolimits\!\left(z\mathop{\sin\/}\nolimits\theta\right)=2\sum_{% {k=0}}^{\infty}\mathop{J_{{2k+1}}\/}\nolimits\!\left(z\right)\mathop{\sin\/}% \nolimits\!\left((2k+1)\theta\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, : nonnegative integer and : complex variable
A&S Ref:: 9.1.42, 9.1.43
Referenced by:: §10.12
Permalink:: http://dlmf.nist.gov/10.12.E2
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10.12.3

$\mathop{\cos\/}\nolimits\!\left(z\mathop{\cos\/}\nolimits\theta\right)=\mathop% {J_{{0}}\/}\nolimits\!\left(z\right)+2\sum_{{k=1}}^{\infty}(-1)^{k}\mathop{J_{% {2k}}\/}\nolimits\!\left(z\right)\mathop{\cos\/}\nolimits\!\left(2k\theta% \right),$

$\mathop{\sin\/}\nolimits\!\left(z\mathop{\cos\/}\nolimits\theta\right)=2\sum_{% {k=0}}^{\infty}(-1)^{k}\mathop{J_{{2k+1}}\/}\nolimits\!\left(z\right)\mathop{% \cos\/}\nolimits\!\left((2k+1)\theta\right).$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, : nonnegative integer and : complex variable
A&S Ref:: 9.1.44, 9.1.45
Permalink:: http://dlmf.nist.gov/10.12.E3
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10.12.4 $1=\mathop{J_{{0}}\/}\nolimits\!\left(z\right)+2\!\mathop{J_{{2}}\/}\nolimits\!% \left(z\right)+2\!\mathop{J_{{4}}\/}\nolimits\!\left(z\right)+2\!\mathop{J_{{6% }}\/}\nolimits\!\left(z\right)+\cdots,$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind and : complex variable
A&S Ref:: 9.1.46
Referenced by:: §10.74(iv)
Permalink:: http://dlmf.nist.gov/10.12.E4
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10.12.5

$\mathop{\cos\/}\nolimits z=\mathop{J_{{0}}\/}\nolimits\!\left(z\right)-2\!% \mathop{J_{{2}}\/}\nolimits\!\left(z\right)+2\!\mathop{J_{{4}}\/}\nolimits\!% \left(z\right)-2\!\mathop{J_{{6}}\/}\nolimits\!\left(z\right)+\cdots,$

$\mathop{\sin\/}\nolimits z=2\!\mathop{J_{{1}}\/}\nolimits\!\left(z\right)-2\!% \mathop{J_{{3}}\/}\nolimits\!\left(z\right)+2\!\mathop{J_{{5}}\/}\nolimits\!% \left(z\right)-\cdots,$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function and : complex variable
A&S Ref:: 9.1.47, 9.1.48
Permalink:: http://dlmf.nist.gov/10.12.E5
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10.12.6

$\tfrac{1}{2}z\mathop{\cos\/}\nolimits z=\mathop{J_{{1}}\/}\nolimits\!\left(z% \right)-9\!\mathop{J_{{3}}\/}\nolimits\!\left(z\right)+25\!\mathop{J_{{5}}\/}% \nolimits\!\left(z\right)-49\!\mathop{J_{{7}}\/}\nolimits\!\left(z\right)+\cdots,$

$\tfrac{1}{2}z\mathop{\sin\/}\nolimits z=4\!\mathop{J_{{2}}\/}\nolimits\!\left(% z\right)-16\!\mathop{J_{{4}}\/}\nolimits\!\left(z\right)+36\!\mathop{J_{{6}}\/% }\nolimits\!\left(z\right)-\cdots.$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function and : complex variable
Referenced by:: §10.12, ¶ ‣ §10.23(iii)
Permalink:: http://dlmf.nist.gov/10.12.E6
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