§10.30 Limiting Forms

Referenced by:: §10.25(iii), §10.28
Permalink:: http://dlmf.nist.gov/10.30

§10.30(i) $z\to 0$
§10.30(ii) $z\to\infty$

§10.30(i) $z\to 0$

Notes:: For (10.30.1) use (10.25.2). For (10.30.2) and (10.30.3) use (10.27.4) when $\nu$ is not an integer; (10.27.3), (10.31.1) otherwise.
Keywords:: modified Bessel functions
Referenced by:: §11.13(iv)
Permalink:: http://dlmf.nist.gov/10.30.i

When $\nu$ is fixed and $z\to 0$ ,

10.30.1 $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)\sim(\tfrac{1}{2}z)^{\nu}/\mathop{\Gamma\/}\nolimits\!\left(\nu+1\right),$ $\nu\neq-1,-2,-3,\dots$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\sim$ : asymptotic equality, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.6.7
Referenced by:: §10.30(i)
Permalink:: http://dlmf.nist.gov/10.30.E1
Encodings:: TeX, pMML, png

10.30.2 $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)\sim\tfrac{1}{2}\mathop{\Gamma\/}\nolimits\!\left(\nu\right)(\tfrac{1}{2}z)^{{-\nu}},$ $\realpart{\nu}>0$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\realpart{}$ : real part, $\sim$ : asymptotic equality, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.6.9
Referenced by:: §10.30(i)
Permalink:: http://dlmf.nist.gov/10.30.E2
Encodings:: TeX, pMML, png

10.30.3 $\mathop{K_{{0}}\/}\nolimits\!\left(z\right)\sim-\mathop{\ln\/}\nolimits z.$

Symbols:: $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\sim$ : asymptotic equality and $z$ : complex variable
A&S Ref:: 9.6.8
Referenced by:: §10.30(i)
Permalink:: http://dlmf.nist.gov/10.30.E3
Encodings:: TeX, pMML, png

For $\mathop{K_{{\nu}}\/}\nolimits\!\left(x\right)$ , when $\nu$ is purely imaginary and $x\to 0+$ , see (10.45.2) and (10.45.7).

§10.30(ii) $z\to\infty$

Notes:: Use (10.40.1) and (10.34.1) with $m=\pm 1$ .
Referenced by:: §11.13(iv), §33.21(i)
Permalink:: http://dlmf.nist.gov/10.30.ii

When $\nu$ is fixed and $z\to\infty$ ,

10.30.4 $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)\sim e^{z}/\sqrt{2\pi z},$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{2}\pi-\delta$ ,

Symbols:: $e$ : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $z$ : complex variable, $\nu$ : complex parameter and $\delta$ : small positive constant
Permalink:: http://dlmf.nist.gov/10.30.E4
Encodings:: TeX, pMML, png

10.30.5 $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)\sim e^{{\pm(\nu+\frac{1}{2})\pi i}}e^{{-z}}/\sqrt{2\pi z},$ $\tfrac{1}{2}\pi+\delta\leq\pm\mathop{\mathrm{ph}\/}\nolimits z\leq\tfrac{3}{2}\pi-\delta$ .

Symbols:: $e$ : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $z$ : complex variable, $\nu$ : complex parameter and $\delta$ : small positive constant
Permalink:: http://dlmf.nist.gov/10.30.E5
Encodings:: TeX, pMML, png

For $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ see (10.25.3).

§10.30 Limiting Forms

Contents

§10.30(i)

§10.30(ii)

§10.30(i) $z\to 0$

§10.30(ii) $z\to\infty$