§10.7 Limiting Forms

Keywords:: Bessel functions, Hankel functions
Referenced by:: §10.2(iii), §10.5
Permalink:: http://dlmf.nist.gov/10.7

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

§10.7(i) $z\to 0$

Notes:: For (10.7.1) and (10.7.3) use (10.2.2) and (10.8.2). For (10.7.2) use (10.4.3) and (10.7.1). For (10.7.4) and (10.7.5) use (10.2.3) and (10.7.3) when $\nu$ is not an integer; (10.4.1), (10.8.1) otherwise. For (10.7.6) use (10.2.3) and (10.7.3). For (10.7.7) use (10.4.3), (10.7.3), and (10.7.4).
Referenced by:: §10.18(i), §11.13(iv)
Permalink:: http://dlmf.nist.gov/10.7.i

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

10.7.1

$\mathop{J_{{0}}\/}\nolimits\!\left(z\right)\to 1,$

$\mathop{Y_{{0}}\/}\nolimits\!\left(z\right)\sim(2/\pi)\mathop{\ln\/}\nolimits z,$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\sim$ : asymptotic equality and $z$ : complex variable
A&S Ref:: 9.1.8
Referenced by:: §10.7(i)
Permalink:: http://dlmf.nist.gov/10.7.E1
Encodings:: TeX, TeX, pMML, pMML, png, png

10.7.2 $\mathop{{H^{{(1)}}_{{0}}}\/}\nolimits\!\left(z\right)\sim-\mathop{{H^{{(2)}}_{{0}}}\/}\nolimits\!\left(z\right)\sim(2i/\pi)\mathop{\ln\/}\nolimits z,$

Symbols:: $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\sim$ : asymptotic equality and $z$ : complex variable
A&S Ref:: 9.1.8
Referenced by:: §10.7(i)
Permalink:: http://dlmf.nist.gov/10.7.E2
Encodings:: TeX, pMML, png

10.7.3 $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)\sim(\tfrac{1}{2}z)^{\nu}/\mathop{\Gamma\/}\nolimits\!\left(\nu+1\right),$ $\nu\neq-1,-2,-3,\dots$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\sim$ : asymptotic equality, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.7
Referenced by:: §10.7(i), §10.7(i)
Permalink:: http://dlmf.nist.gov/10.7.E3
Encodings:: TeX, pMML, png

10.7.4 $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)\sim-(1/\pi)\mathop{\Gamma\/}\nolimits\!\left(\nu\right)(\tfrac{1}{2}z)^{{-\nu}},$ $\realpart{\nu}>0$ or $\nu=-\tfrac{1}{2},-\tfrac{3}{2},-\tfrac{5}{2},\ldots$ ,

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\realpart{}$ : real part, $\sim$ : asymptotic equality, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.9
Referenced by:: §10.7(i)
Permalink:: http://dlmf.nist.gov/10.7.E4
Encodings:: TeX, pMML, png

10.7.5 $\mathop{Y_{{-\nu}}\/}\nolimits\!\left(z\right)\sim-(1/\pi)\mathop{\cos\/}\nolimits(\nu\pi)\mathop{\Gamma\/}\nolimits\!\left(\nu\right)(\tfrac{1}{2}z)^{{-\nu}},$ $\realpart{\nu}>0$ , $\nu\neq\tfrac{1}{2},\tfrac{3}{2},\tfrac{5}{2},\ldots$ ,

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\realpart{}$ : real part, $\sim$ : asymptotic equality, $z$ : complex variable and $\nu$ : complex parameter
Referenced by:: §10.7(i)
Permalink:: http://dlmf.nist.gov/10.7.E5
Encodings:: TeX, pMML, png

10.7.6 $\mathop{Y_{{i\nu}}\/}\nolimits\!\left(z\right)=\frac{i\mathop{\mathrm{csch}\/}\nolimits(\nu\pi)}{\mathop{\Gamma\/}\nolimits\!\left(1-i\nu\right)}(\tfrac{1}{2}z)^{{-i\nu}}-\frac{i\mathop{\coth\/}\nolimits(\nu\pi)}{\mathop{\Gamma\/}\nolimits\!\left(1+i\nu\right)}(\tfrac{1}{2}z)^{{i\nu}}+e^{{|\nu\mathop{\mathrm{ph}\/}\nolimits z|}}\mathop{o\/}\nolimits\!\left(1\right),$ $\nu\in\Real$ and $\nu\neq 0$ .

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\in$ : element of, $e$ : base of exponential function, $\mathop{\mathrm{csch}\/}\nolimits z$ : hyperbolic cosecant function, $\mathop{\coth\/}\nolimits z$ : hyperbolic cotangent function, $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\Real$ : real line (excluding infinity), $z$ : complex variable and $\nu$ : complex parameter
Referenced by:: §10.7(i), §10.7(i)
Permalink:: http://dlmf.nist.gov/10.7.E6
Encodings:: TeX, pMML, png

§10.7(ii) $z\to\infty$

Notes:: See (10.17.3) and (10.17.4).
Referenced by:: §11.13(iv), §33.21(i)
Permalink:: http://dlmf.nist.gov/10.7.ii

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

10.7.8

$\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)=\sqrt{2/(\pi z)}\left(\mathop{\cos\/}\nolimits\!\left(z-\tfrac{1}{2}\nu\pi-\tfrac{1}{4}\pi\right)+e^{{|\imagpart{z}|}}\mathop{o\/}\nolimits\!\left(1\right)\right),$

$\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)=\sqrt{2/(\pi z)}\left(\mathop{\sin\/}\nolimits\!\left(z-\tfrac{1}{2}\nu\pi-\tfrac{1}{4}\pi\right)+e^{{|\imagpart{z}|}}\mathop{o\/}\nolimits\!\left(1\right)\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi-\delta(<\pi)$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $e$ : base of exponential function, $\imagpart{}$ : imaginary part, $\mathop{o\/}\nolimits\!\left(x\right)$ : order less than, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\mathop{\sin\/}\nolimits z$ : sine function, $z$ : complex variable, $\nu$ : complex parameter and $\delta$ : small positive constant
A&S Ref:: 9.2.1, 9.2.2
Permalink:: http://dlmf.nist.gov/10.7.E8
Encodings:: TeX, TeX, pMML, pMML, png, png

For the corresponding results for $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ and $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ see (10.2.5) and (10.2.6).

§10.7 Limiting Forms

Contents

§10.7(i)

§10.7(ii)

§10.7(i) $z\to 0$

§10.7(ii) $z\to\infty$