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21: 6.12 Asymptotic Expansions

… ►When $|\mathrm{ph}z|\le \frac{1}{2}\pi$ the remainder is bounded in magnitude by the first neglected term, and has the same sign when $\mathrm{ph}z=0$. When the remainder term is bounded in magnitude by $\csc \left(|\mathrm{ph}z|\right)$ times the first neglected term. … ►When $|\mathrm{ph}z|\le \frac{1}{4}\pi$, these remainders are bounded in magnitude by the first neglected terms in (6.12.3) and (6.12.4), respectively, and have the same signs as these terms when $\mathrm{ph}z=0$. When the remainders are bounded in magnitude by $\csc \left(2|\mathrm{ph}z|\right)$ times the first neglected terms. ►For other phase ranges use (6.4.6) and (6.4.7). …

22: 10.40 Asymptotic Expansions for Large Argument

… ►

10.40.3 $$I_{\nu }^{\prime }\left(z\right)\sim \frac{{\mathrm{e}}^z}{{(2\pi z)}^{\frac{1}{2}}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{b_k(\nu )}{z^k},$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$,

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant and $b_k(\nu )$: polynomial coefficient

A&S Ref:

9.7.3

Referenced by:

§10.40(i)

Permalink:

http://dlmf.nist.gov/10.40.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(i), §10.40 and Ch.10

►

10.40.4 $$K_{\nu }^{\prime }\left(z\right)\sim -{\left(\frac{\pi }{2z}\right)}^{\frac{1}{2}}{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{b_k(\nu )}{z^k},$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$.

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant and $b_k(\nu )$: polynomial coefficient

A&S Ref:

9.7.4

Referenced by:

§10.40(i), §10.40(i), §10.67(i)

Permalink:

http://dlmf.nist.gov/10.40.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(i), §10.40 and Ch.10

►Corresponding expansions for $I_{\nu }\left(z\right)$, $K_{\nu }\left(z\right)$, $I_{\nu }^{\prime }\left(z\right)$, and $K_{\nu }^{\prime }\left(z\right)$ for other ranges of $\mathrm{ph}z$ are obtainable by combining (10.34.3), (10.34.4), (10.34.6), and their differentiated forms, with (10.40.2) and (10.40.4). … ►as $z\to \mathrm{\infty }$ in $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$. … ►as $z\to \mathrm{\infty }$ in $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$. …

23: 10.17 Asymptotic Expansions for Large Argument

… ►

10.17.3 $$J_{\nu }\left(z\right)\sim {\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}\left(\cos \omega \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{a_{2k}(\nu )}{z^{2k}}-\sin \omega \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{a_{2k+1}(\nu )}{z^{2k+1}}\right),$$ $|\mathrm{ph}z|\le \pi -\delta$,

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{ph}$: phase, $\sin z$: sine function, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant, $a_k(\nu )$: polynomial coefficient and $\omega$

A&S Ref:

9.2.5

Referenced by:

§10.17(i), §10.17(iii), §10.17(iv), §10.18(iii), §10.24, §10.49(i), §10.61(v), §10.7(ii)

Permalink:

http://dlmf.nist.gov/10.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.17(i), §10.17 and Ch.10

… ►

10.17.7 $$z^{\frac{1}{2}}=\exp \left(\frac{1}{2}\ln |z|+\frac{1}{2}\mathrm{i}\mathrm{ph}z\right).$$

ⓘ

Symbols:

$\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.17.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.17(i), §10.17 and Ch.10

►Corresponding expansions for other ranges of $\mathrm{ph}z$ can be obtained by combining (10.17.3), (10.17.5), (10.17.6) with the continuation formulas (10.11.1), (10.11.3), (10.11.4) (or (10.11.7), (10.11.8)), and also the connection formula given by the second of (10.4.4). … ►

10.17.15

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, ${𝒱}_{a,b}\left(f\right)$: total variation, $z$: complex variable and $\chi (\mathrm{\ell })$

Referenced by:

§10.17(iv)

Permalink:

http://dlmf.nist.gov/10.17.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §10.17(iv), §10.17 and Ch.10

… ►

10.17.18 $$R_{m,\mathrm{\ell }}^{\pm }(\nu ,z)=O\left({\mathrm{e}}^{-2|z|}{z}^{-m}\right),$$ $|\mathrm{ph}\left(z{\mathrm{e}}^{\mp \frac{1}{2}\pi \mathrm{i}}\right)|\le \pi$.

ⓘ

Defines:

$R_{\mathrm{\ell }}^{\pm }(\nu ,z)$: remainder (locally)

Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $m$: integer, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.17.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §10.17(v), §10.17 and Ch.10

…

24: 12.5 Integral Representations

… ►

12.5.2 $$U(a,z)=\frac{z{\mathrm{e}}^{-\frac{1}{4}{z}^2}}{\mathrm{\Gamma }\left(\frac{1}{4}+\frac{1}{2}a\right)}{\int }_0^{\mathrm{\infty }}{t}^{\frac{1}{2}a-\frac{3}{4}}{\mathrm{e}}^{-t}{\left(z^2+2t\right)}^{-\frac{1}{2}a-\frac{3}{4}}dt,$$ , $\mathrm{\Re }a>-\frac{1}{2}$ ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.5.11 (modification of)

Permalink:

http://dlmf.nist.gov/12.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.5(i), §12.5 and Ch.12

►

12.5.3 $$U(a,z)=\frac{{\mathrm{e}}^{-\frac{1}{4}{z}^2}}{\mathrm{\Gamma }\left(\frac{3}{4}+\frac{1}{2}a\right)}{\int }_0^{\mathrm{\infty }}{t}^{\frac{1}{2}a-\frac{1}{4}}{\mathrm{e}}^{-t}{\left(z^2+2t\right)}^{-\frac{1}{2}a-\frac{1}{4}}dt,$$ , $\mathrm{\Re }a>-\frac{3}{2}$ ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.5.12 (modification of)

Permalink:

http://dlmf.nist.gov/12.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.5(i), §12.5 and Ch.12

… ►

12.5.5 $$U(a,z)=\frac{\mathrm{\Gamma }\left(\frac{1}{2}-a\right)}{2\pi \mathrm{i}}{\mathrm{e}}^{-\frac{1}{4}{z}^2}{\int }_{-\mathrm{\infty }}^{(0+)}{\mathrm{e}}^{zt-\frac{1}{2}{t}^2}{t}^{a-\frac{1}{2}}dt,$$ $a\ne \frac{1}{2},\frac{3}{2},\frac{5}{2},\mathrm{\dots }$, .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.5.1 (modification of)

Permalink:

http://dlmf.nist.gov/12.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §12.5(ii), §12.5 and Ch.12

… ►

12.5.6 $$U(a,z)=\frac{{\mathrm{e}}^{\frac{1}{4}{z}^2}}{\mathrm{i}\sqrt{2\pi }}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}{\mathrm{e}}^{-zt+\frac{1}{2}{t}^2}{t}^{-a-\frac{1}{2}}dt,$$ , $c>0$ ,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.5.4 (modification of)

Referenced by:

§12.8(ii)

Permalink:

http://dlmf.nist.gov/12.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §12.5(ii), §12.5 and Ch.12

►

12.5.7 $$V(a,z)=\frac{{\mathrm{e}}^{-\frac{1}{4}{z}^2}}{2\pi }\left({\int }_{-\mathrm{i}c-\mathrm{\infty }}^{-\mathrm{i}c+\mathrm{\infty }}+{\int }_{\mathrm{i}c-\mathrm{\infty }}^{\mathrm{i}c+\mathrm{\infty }}\right){\mathrm{e}}^{zt-\frac{1}{2}{t}^2}{t}^{a-\frac{1}{2}}dt,$$ , $c>0$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $V(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable and $a$: real or complex parameter

Referenced by:

§12.8(ii)

Permalink:

http://dlmf.nist.gov/12.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §12.5(ii), §12.5 and Ch.12

…

25: 10.32 Integral Representations

… ►

10.32.9 $$K_{\nu }\left(z\right)={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-z\cosh t}\cosh \left(\nu t\right)dt,$$ .

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.24

Referenced by:

§10.43(iii)

Permalink:

http://dlmf.nist.gov/10.32.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.32(i), §10.32 and Ch.10

►

10.32.10 $$K_{\nu }\left(z\right)=\frac{1}{2}{(\frac{1}{2}z)}^{\nu }{\int }_0^{\mathrm{\infty }}\exp \left(-t-\frac{z^2}{4t}\right)\frac{dt}{t^{\nu +1}},$$ .

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§6.7(iii), §7.14(i)

Permalink:

http://dlmf.nist.gov/10.32.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §10.32(i), §10.32 and Ch.10

… ►

10.32.12 $$I_{\nu }\left(z\right)=\frac{1}{2\pi \mathrm{i}}{\int }_{\mathrm{\infty }-\mathrm{i}\pi }^{\mathrm{\infty }+\mathrm{i}\pi }{\mathrm{e}}^{z\cosh t-\nu t}dt,$$ .

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.32.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §10.32(ii), §10.32 and Ch.10

… ►

10.32.17 $$K_{\mu }\left(z\right)K_{\nu }\left(z\right)=2{\int }_0^{\mathrm{\infty }}{K}_{\mu \pm \nu }\left(2z\cosh t\right)\cosh \left((\mu \mp \nu )t\right)dt,$$ .

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.32.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §10.32(iii), §10.32 and Ch.10

►

10.32.18 $$K_{\nu }\left(z\right)K_{\nu }\left(\zeta \right)=\frac{1}{2}{\int }_0^{\mathrm{\infty }}\exp \left(-\frac{t}{2}-\frac{z^2+{\zeta }^2}{2t}\right)K_{\nu }\left(\frac{z\zeta }{t}\right)\frac{dt}{t},$$ , , .

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.32.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §10.32(iii), §10.32 and Ch.10

…

26: 10.54 Integral Representations

… ►

10.54.3 $${𝗄}_n\left(z\right)=\frac{\pi }{2}{\int }_1^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{P}_n\left(t\right)dt,$$

ⓘ

Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase, ${𝗄}_n\left(z\right)$: modified spherical Bessel function, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.54.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.54 and Ch.10

►

10.54.4 $${𝗃}_n\left(z\right)=\frac{{(-\mathrm{i})}^{n+1}}{2\pi }{\int }_{\mathrm{i}\mathrm{\infty }}^{(-1+,1+)}{\mathrm{e}}^{\mathrm{i}zt}{Q}_n\left(t\right)dt,$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $(a,b)$: open interval, $\mathrm{ph}$: phase, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.54.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.54 and Ch.10

… ►

${𝗁}_n^{(2)}\left(z\right)={\displaystyle \frac{{(-\mathrm{i})}^{n+1}}{\pi }}{\displaystyle {\int }_{\mathrm{i}\mathrm{\infty }}^{(-1+)}}{\mathrm{e}}^{\mathrm{i}zt}{Q}_n\left(t\right)dt,$

…

27: 13.4 Integral Representations

… ►

13.4.4 $$U(a,b,z)=\frac{1}{\mathrm{\Gamma }\left(a\right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{a-1}{(1+t)}^{b-a-1}dt,$$ $\mathrm{\Re }a>0$, ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

13.2.5

Referenced by:

§13.10(v), §13.29(iii), §7.11, (9.5.8)

Permalink:

http://dlmf.nist.gov/13.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(i), §13.4 and Ch.13

►

13.4.5 $$U(a,b,z)=\frac{z^{1-a}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(1+a-b\right)}{\int }_0^{\mathrm{\infty }}\frac{U(b-a,b,t){\mathrm{e}}^{-t}{t}^{a-1}}{t+z}dt,$$ , $\mathrm{\Re }a>\max (\mathrm{\Re }b-1,0)$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§13.10(vi), §13.4(i), (9.10.18)

Permalink:

http://dlmf.nist.gov/13.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(i), §13.4 and Ch.13

… ►

13.4.8 $$U(a,b,z)=z^{c-a}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{c-1}{}_2{}^{}𝐅_1^{}(a,a-b+1;c;-t)dt,$$ ,

ⓘ

Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $\int$: integral, $\mathrm{ph}$: phase, $z$: complex variable and $c$: arbitrary

Permalink:

http://dlmf.nist.gov/13.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(i), §13.4 and Ch.13

… ►

13.4.13 $$𝐌(a,b,z)=\frac{z^{1-b}}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+,1+)}{\mathrm{e}}^{zt}{t}^{-b}{\left(1-\frac{1}{t}\right)}^{-a}dt,$$ .

ⓘ

Symbols:

$𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $(a,b)$: open interval, $\mathrm{ph}$: phase and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.4.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(ii), §13.4 and Ch.13

… ►At this point the fractional powers are determined by $\mathrm{ph}t=\pi$ and $\mathrm{ph}\left(1+t\right)=0$. …

28: 10.27 Connection Formulas

… ►

10.27.6 $$I_{\nu }\left(z\right)={\mathrm{e}}^{\mp \nu \pi \mathrm{i}/2}{J}_{\nu }\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}/2}\right),$$ $-\pi \le \pm \mathrm{ph}z\le \frac{1}{2}\pi$,

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.3 (modified)

Referenced by:

§10.26(ii), §10.27, §10.27, §10.31, §10.33, §10.35, §10.39, §10.41(i), §10.42, §10.43(ii), §10.43(iv), §10.44(i), §10.44(ii), §10.44(iii), §10.47(iv), §10.67(i), §11.4(i)

Permalink:

http://dlmf.nist.gov/10.27.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.27 and Ch.10

►

10.27.7 $$I_{\nu }\left(z\right)=\frac{1}{2}{\mathrm{e}}^{\mp \nu \pi \mathrm{i}/2}\left(H_{\nu }^{(1)}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}/2}\right)+H_{\nu }^{(2)}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}/2}\right)\right),$$ $-\pi \le \pm \mathrm{ph}z\le \frac{1}{2}\pi$.

ⓘ

Symbols:

$H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.27.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.27 and Ch.10

… ►

10.27.9 $$\pi \mathrm{i}{J}_{\nu }\left(z\right)={\mathrm{e}}^{-\nu \pi \mathrm{i}/2}{K}_{\nu }\left(z{\mathrm{e}}^{-\pi \mathrm{i}/2}\right)-{\mathrm{e}}^{\nu \pi \mathrm{i}/2}{K}_{\nu }\left(z{\mathrm{e}}^{\pi \mathrm{i}/2}\right),$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi$.

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.67(i)

Permalink:

http://dlmf.nist.gov/10.27.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.27 and Ch.10

►

10.27.10 $$-\pi Y_{\nu }\left(z\right)={\mathrm{e}}^{-\nu \pi \mathrm{i}/2}{K}_{\nu }\left(z{\mathrm{e}}^{-\pi \mathrm{i}/2}\right)+{\mathrm{e}}^{\nu \pi \mathrm{i}/2}{K}_{\nu }\left(z{\mathrm{e}}^{\pi \mathrm{i}/2}\right),$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi$.

ⓘ

Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.43(iv)

Permalink:

http://dlmf.nist.gov/10.27.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §10.27 and Ch.10

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10.27.11 $$Y_{\nu }\left(z\right)={\mathrm{e}}^{\pm (\nu +1)\pi \mathrm{i}/2}{I}_{\nu }\left(z{\mathrm{e}}^{\mp \pi \mathrm{i}/2}\right)-(2/\pi ){\mathrm{e}}^{\mp \nu \pi \mathrm{i}/2}{K}_{\nu }\left(z{\mathrm{e}}^{\mp \pi \mathrm{i}/2}\right),$$ $-\frac{1}{2}\pi \le \pm \mathrm{ph}z\le \pi$.

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Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.5 (modified)

Referenced by:

§11.6(i)

Permalink:

http://dlmf.nist.gov/10.27.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §10.27 and Ch.10

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29: 8.9 Continued Fractions

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8.9.2 $$z^{-a}{\mathrm{e}}^z\mathrm{\Gamma }(a,z)=\frac{z^{-1}}{1+}\frac{(1-a)z^{-1}}{1+}\frac{z^{-1}}{1+}\frac{(2-a)z^{-1}}{1+}\frac{2z^{-1}}{1+}\frac{(3-a)z^{-1}}{1+}\frac{3z^{-1}}{1+}\mathrm{\cdots },$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\mathrm{ph}$: phase, $z$: complex variable and $a$: parameter

Referenced by:

§8.19(vii)

Permalink:

http://dlmf.nist.gov/8.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.9 and Ch.8

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30: 9.7 Asymptotic Expansions

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9.7.5 $$\mathrm{Ai}\left(z\right)\sim \frac{{\mathrm{e}}^{-\zeta }}{2\sqrt{\pi }{z}^{1/4}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{u_k}{{\zeta }^k},$$ $|\mathrm{ph}z|\le \pi -\delta$,

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $u_s$: expansion coefficient

Source:

Olver (1997b, (1.07), pp. 392 and 413)

Referenced by:

(9.10.4), (9.10.6), §9.7(iii), §9.7(iv), §9.7(iv), §9.7(v), Erratum (V1.0.17) for Subsection 9.7(iv)

Permalink:

http://dlmf.nist.gov/9.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(ii), §9.7 and Ch.9

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9.7.6 $${\mathrm{Ai}}^{\prime }\left(z\right)\sim -\frac{z^{1/4}{\mathrm{e}}^{-\zeta }}{2\sqrt{\pi }}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{v_k}{{\zeta }^k},$$ $|\mathrm{ph}z|\le \pi -\delta$,

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $v_s$: expansion coefficient

Source:

Olver (1997b, (1.07), pp. 392 and 413)

Referenced by:

§9.7(iii), §9.7(iv), §9.7(iv), §9.7(v), Erratum (V1.0.17) for Subsection 9.7(iv)

Permalink:

http://dlmf.nist.gov/9.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(ii), §9.7 and Ch.9

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9.7.7 $$\mathrm{Bi}\left(z\right)\sim \frac{{\mathrm{e}}^{\zeta }}{\sqrt{\pi }{z}^{1/4}}\sum _{k=0}^{\mathrm{\infty }}\frac{u_k}{{\zeta }^k},$$ $|\mathrm{ph}z|\le \frac{1}{3}\pi -\delta$,

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $u_s$: expansion coefficient

Source:

Olver (1997b, (1.16), pp. 393 and 414)

Referenced by:

(9.10.5), (9.10.7), §9.7(iii), §9.7(iii), §9.7(iv), §9.7(v), Erratum (V1.0.17) for Subsection 9.7(iii)

Permalink:

http://dlmf.nist.gov/9.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(ii), §9.7 and Ch.9

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9.7.8 $${\mathrm{Bi}}^{\prime }\left(z\right)\sim \frac{z^{1/4}{\mathrm{e}}^{\zeta }}{\sqrt{\pi }}\sum _{k=0}^{\mathrm{\infty }}\frac{v_k}{{\zeta }^k},$$ $|\mathrm{ph}z|\le \frac{1}{3}\pi -\delta$.

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $v_s$: expansion coefficient

Source:

Olver (1997b, (1.16), pp. 393 and 414)

Referenced by:

§9.7(iii), §9.7(iii), Erratum (V1.0.17) for Subsection 9.7(iii)

Permalink:

http://dlmf.nist.gov/9.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(ii), §9.7 and Ch.9

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9.7.17

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cos z$: cosine function, $\exp z$: exponential function, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $z$: complex variable, $\zeta$: change of variable, $\chi (x)$: function, $n$: index and $\sigma$: real variable

Proof sketch:

Derivable from (9.6.1)–(9.6.5) and Nemes (2017b, (26) and Propositions B.1, B.3, B.4).

Referenced by:

§9.7(iv), Erratum (V1.0.11) for Equation (9.7.17)

Permalink:

http://dlmf.nist.gov/9.7.E17

Encodings:

pMML, png, TeX

Modification (effective with 1.0.17):

The bounds have been sharpened for $|\mathrm{ph}z|\le \frac{1}{3}\pi$, from $2\exp \left({\displaystyle \frac{\sigma }{36|\zeta |}}\right)$ to $1$; for $\frac{1}{3}\pi \le |\mathrm{ph}z|\le \frac{2}{3}\pi$, from $2\chi (n)\exp \left({\displaystyle \frac{\sigma \pi }{72|\zeta |}}\right)$ to $\min (|\csc \left(\mathrm{ph}\zeta \right)|,\chi (n+\sigma )+1)$; and for from ${\displaystyle \frac{4\chi (n)}{{|\cos \left(\mathrm{ph}\zeta \right)|}^n}}\exp \left({\displaystyle \frac{\sigma \pi }{36|\mathrm{\Re }\zeta |}}\right)$ to $\frac{\sqrt{2\pi (n+\sigma )}}{{|\cos \left(\mathrm{ph}\zeta \right)|}^{n+\sigma }}+\chi (n+\sigma )+1$.

Errata (effective with 1.0.11):

Originally the constraint condition was written incorrectly as $\frac{2}{3}\pi \le |\mathrm{ph}z|\le \pi$. Also, the equation was reformatted to display the constraints in the equation instead of in the text.

Reported 2014-11-05 by Gergő Nemes

See also:

Annotations for §9.7(iv), §9.7 and Ch.9

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