§11.6 Asymptotic Expansions

§11.6(i) Large $|z|$, Fixed $\nu$

 11.6.1 $\mathop{\mathbf{K}_{\nu}\/}\nolimits\!\left(z\right)\sim\frac{1}{\pi}\sum_{k=0% }^{\infty}\frac{\mathop{\Gamma\/}\nolimits\!\left(k+\tfrac{1}{2}\right)(\tfrac% {1}{2}z)^{\nu-2k-1}}{\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{1}{2}-k% \right)},$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi-\delta$,

where $\delta$ is an arbitrary small positive constant. If the series on the right-hand side of (11.6.1) is truncated after $m(\geq 0)$ terms, then the remainder term $R_{m}(z)$ is $\mathop{O\/}\nolimits\!\left(z^{\nu-2m-1}\right)$. If $\nu$ is real, $z$ is positive, and $m+\tfrac{1}{2}-\nu\geq 0$, then $R_{m}(z)$ is of the same sign and numerically less than the first neglected term.

 11.6.2 $\mathop{\mathbf{M}_{\nu}\/}\nolimits\!\left(z\right)\sim\frac{1}{\pi}\sum_{k=0% }^{\infty}(-1)^{k+1}\frac{\mathop{\Gamma\/}\nolimits\!\left(k+\tfrac{1}{2}% \right)(\tfrac{1}{2}z)^{\nu-2k-1}}{\mathop{\Gamma\/}\nolimits\!\left(\nu+% \tfrac{1}{2}-k\right)},$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{2}\pi-\delta$.

For re-expansions of the remainder terms in (11.6.1) and (11.6.2), see Dingle (1973, p. 445).

For the corresponding expansions for $\mathop{\mathbf{H}_{\nu}\/}\nolimits\!\left(z\right)$ and $\mathop{\mathbf{L}_{\nu}\/}\nolimits\!\left(z\right)$ combine (11.6.1), (11.6.2) with (11.2.5), (11.2.6), (10.17.4), and (10.40.1).

 11.6.3 $\int_{0}^{z}\mathop{\mathbf{K}_{0}\/}\nolimits\!\left(t\right)\mathrm{d}t-% \frac{2}{\pi}(\mathop{\ln\/}\nolimits\!\left(2z\right)+\gamma)\sim\frac{2}{\pi% }\sum_{k=1}^{\infty}(-1)^{k+1}\frac{(2k)!(2k-1)!}{(k!)^{2}(2z)^{2k}},$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi-\delta$,
 11.6.4 $\int_{0}^{z}\mathop{\mathbf{M}_{0}\/}\nolimits\!\left(t\right)\mathrm{d}t+% \frac{2}{\pi}(\mathop{\ln\/}\nolimits\!\left(2z\right)+\gamma)\sim\frac{2}{\pi% }\sum_{k=1}^{\infty}\frac{(2k)!(2k-1)!}{(k!)^{2}(2z)^{2k}},$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{2}\pi-\delta$,

where $\gamma$ is Euler’s constant (§5.2(ii)).

§11.6(ii) Large $|\nu|$, Fixed $z$

 11.6.5 $\mathop{\mathbf{H}_{\nu}\/}\nolimits\!\left(z\right),\mathop{\mathbf{L}_{\nu}% \/}\nolimits\!\left(z\right)\sim\frac{z}{\pi\nu\sqrt{2}}\left(\frac{ez}{2\nu}% \right)^{\nu},$ $|\mathop{\mathrm{ph}\/}\nolimits\nu|\leq\pi-\delta$.

More fully, the series (11.2.1) and (11.2.2) can be regarded as generalized asymptotic expansions (§2.1(v)).

§11.6(iii) Large $|\nu|$, Fixed $z/\nu$

For fixed $\lambda(>1)$

 11.6.6 $\mathop{\mathbf{K}_{\nu}\/}\nolimits\!\left(\lambda\nu\right)\sim\frac{(\tfrac% {1}{2}\lambda\nu)^{\nu-1}}{\sqrt{\pi}\mathop{\Gamma\/}\nolimits\!\left(\nu+% \tfrac{1}{2}\right)}\sum_{k=0}^{\infty}\frac{k!c_{k}(\lambda)}{\nu^{k}},$ $|\mathop{\mathrm{ph}\/}\nolimits\nu|\leq\tfrac{1}{2}\pi-\delta$,

and for fixed $\lambda$ $(>0)$

 11.6.7 $\mathop{\mathbf{M}_{\nu}\/}\nolimits\!\left(\lambda\nu\right)\sim-\frac{(\frac% {1}{2}\lambda\nu)^{\nu-1}}{\sqrt{\pi}\mathop{\Gamma\/}\nolimits\!\left(\nu+% \frac{1}{2}\right)}\sum_{k=0}^{\infty}\frac{k!c_{k}(i\lambda)}{\nu^{k}},$ $|\mathop{\mathrm{ph}\/}\nolimits\nu|\leq\frac{1}{2}\pi-\delta$.

Here

 11.6.8 $\displaystyle c_{0}(\lambda)$ $\displaystyle=1,$ $\displaystyle c_{1}(\lambda)$ $\displaystyle=2\lambda^{-2},$ $\displaystyle c_{2}(\lambda)$ $\displaystyle=6\lambda^{-4}-\tfrac{1}{2}\lambda^{-2},$ $\displaystyle c_{3}(\lambda)$ $\displaystyle=20\lambda^{-6}-4\lambda^{-4},$ $\displaystyle c_{4}(\lambda)$ $\displaystyle=70\lambda^{-8}-\tfrac{45}{2}\lambda^{-6}+\tfrac{3}{8}\lambda^{-4}.$ Symbols: $\lambda$: parameter and $c_{k}(\lambda)$: expansion function Referenced by: §11.6(iii) Permalink: http://dlmf.nist.gov/11.6.E8 Encodings: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png See also: Annotations for 11.6(iii)

These and higher coefficients $c_{k}(\lambda)$ can be computed via the representations in Nemes (2015b).

For the corresponding result for $\mathop{\mathbf{H}_{\nu}\/}\nolimits\!\left(\lambda\nu\right)$ use (11.2.5) and (10.19.6). See also Watson (1944, p. 336).

For fixed $\lambda$ $(>0)$

 11.6.9 $\mathop{\mathbf{L}_{\nu}\/}\nolimits\!\left(\lambda\nu\right)\sim\mathop{I_{% \nu}\/}\nolimits\!\left(\lambda\nu\right),$ $|\mathop{\mathrm{ph}\/}\nolimits\nu|\leq\tfrac{1}{2}\pi-\delta$,

and for an estimate of the relative error in this approximation see Watson (1944, p. 336).