10 Bessel Functions Modified Bessel Functions 10.34 Analytic Continuation 10.36 Other Differential Equations

§10.35 Generating Function and Associated Series

Notes:: For (10.35.1) replace and in (10.12.1) by and , respectively, and apply (10.27.6). (10.35.2)–(10.35.6) are obtained by setting $t=e^{{i\theta}}$ , $t=-ie^{{i\theta}}$ , together with other straightforward substitutions.
Keywords:: modified Bessel functions
Permalink:: http://dlmf.nist.gov/10.35

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

10.35.1 $e^{{\frac{1}{2}z(t+t^{{-1}})}}=\sum_{{m=-\infty}}^{\infty}t^{m}\mathop{I_{{m}}% \/}\nolimits\!\left(z\right).$

Symbols:: : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, : integer and : complex variable
A&S Ref:: 9.6.33
Referenced by:: §10.35
Permalink:: http://dlmf.nist.gov/10.35.E1
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For $z,\theta\in\Complex$ ,

10.35.2 $e^{{z\mathop{\cos\/}\nolimits\theta}}=\mathop{I_{{0}}\/}\nolimits\!\left(z% \right)+2\sum_{{k=1}}^{\infty}\mathop{I_{{k}}\/}\nolimits\!\left(z\right)% \mathop{\cos\/}\nolimits\!\left(k\theta\right),$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, : nonnegative integer and : complex variable
A&S Ref:: 9.6.34
Referenced by:: §10.35
Permalink:: http://dlmf.nist.gov/10.35.E2
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10.35.3 $e^{{z\mathop{\sin\/}\nolimits\theta}}=\mathop{I_{{0}}\/}\nolimits\!\left(z% \right)+2\sum_{{k=0}}^{\infty}(-1)^{k}\mathop{I_{{2k+1}}\/}\nolimits\!\left(z% \right)\mathop{\sin\/}\nolimits\!\left((2k+1)\theta\right)+2\sum_{{k=1}}^{% \infty}(-1)^{k}\mathop{I_{{2k}}\/}\nolimits\!\left(z\right)\mathop{\cos\/}% \nolimits\!\left(2k\theta\right).$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\sin\/}\nolimits z$ : sine function, : nonnegative integer and : complex variable
A&S Ref:: 9.6.35
Permalink:: http://dlmf.nist.gov/10.35.E3
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10.35.4 $1=\mathop{I_{{0}}\/}\nolimits\!\left(z\right)-2\!\mathop{I_{{2}}\/}\nolimits\!% \left(z\right)+2\!\mathop{I_{{4}}\/}\nolimits\!\left(z\right)-2\!\mathop{I_{{6% }}\/}\nolimits\!\left(z\right)+\cdots,$

Symbols:: $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function and : complex variable
A&S Ref:: 9.6.36
Permalink:: http://dlmf.nist.gov/10.35.E4
Encodings:: TeX, pMML, png

10.35.5 $e^{{\pm z}}=\mathop{I_{{0}}\/}\nolimits\!\left(z\right)\pm 2\!\mathop{I_{{1}}% \/}\nolimits\!\left(z\right)+2\!\mathop{I_{{2}}\/}\nolimits\!\left(z\right)\pm 2% \!\mathop{I_{{3}}\/}\nolimits\!\left(z\right)+\cdots,$

Symbols:: : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function and : complex variable
A&S Ref:: 9.6.37,9.6.38
Permalink:: http://dlmf.nist.gov/10.35.E5
Encodings:: TeX, pMML, png

10.35.6

$\mathop{\cosh\/}\nolimits z=\mathop{I_{{0}}\/}\nolimits\!\left(z\right)+2\!% \mathop{I_{{2}}\/}\nolimits\!\left(z\right)+2\!\mathop{I_{{4}}\/}\nolimits\!% \left(z\right)+2\!\mathop{I_{{6}}\/}\nolimits\!\left(z\right)+\dots,$

$\mathop{\sinh\/}\nolimits z=2\!\mathop{I_{{1}}\/}\nolimits\!\left(z\right)+2\!% \mathop{I_{{3}}\/}\nolimits\!\left(z\right)+2\!\mathop{I_{{5}}\/}\nolimits\!% \left(z\right)+\dots.$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function and : complex variable
A&S Ref:: 9.6.39, 9.6.40
Referenced by:: §10.35
Permalink:: http://dlmf.nist.gov/10.35.E6
Encodings:: TeX, TeX, pMML, pMML, png, png

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