general bases

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11: Bibliography B

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Mathieu functions of general order: Connection formulae, base functions and asymptotic formulae. I–V. Philos. Trans. Roy. Soc. London Ser. A 301, pp. 75–162.

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External Links:

ISSN 0080-4614, FullText, MathReview (J. Meixner), ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

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12: 24.16 Generalizations

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24.16.1 $${\left(\frac{t}{{\mathrm{e}}^t-1}\right)}^{\mathrm{\ell }}{\mathrm{e}}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{B}_n^{(\mathrm{\ell })}\left(x\right)\frac{t^n}{n!},$$ ,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $B_n^{(\mathrm{\ell })}\left(x\right)$: generalized Bernoulli polynomials, $\mathrm{\ell }$: integer, $n$: integer, $t$: real or complex and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §24.16(i), §24.16(i), §24.16 and Ch.24

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24.16.2 $${\left(\frac{2}{{\mathrm{e}}^t+1}\right)}^{\mathrm{\ell }}{\mathrm{e}}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{E}_n^{(\mathrm{\ell })}\left(x\right)\frac{t^n}{n!},$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $E_n^{(\mathrm{\ell })}\left(x\right)$: generalized Euler polynomials, $\mathrm{\ell }$: integer, $n$: integer, $t$: real or complex and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §24.16(i), §24.16(i), §24.16 and Ch.24

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24.16.9 $${\left(\frac{t}{{\mathrm{e}}^t-1}\right)}^x=\sum _{n=0}^{\mathrm{\infty }}{B}_n^{(x)}\frac{t^n}{n!},$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $B_n^{(\mathrm{\ell })}\left(x\right)$: generalized Bernoulli polynomials, $n$: integer, $t$: real or complex and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.16.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.16(i), §24.16(i), §24.16 and Ch.24

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24.16.10 $$\sum _{a=1}^f\frac{\chi (a)t{\mathrm{e}}^{at}}{{\mathrm{e}}^{ft}-1}=\sum _{n=0}^{\mathrm{\infty }}{B}_{n,\chi }\frac{t^n}{n!},$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: integer, $t$: real or complex, $\chi (a)$: Dirichlet character, $f$: conductor and $B_{n,\chi }$: Generalized Bernoulli numbers

Permalink:

http://dlmf.nist.gov/24.16.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §24.16(ii), §24.16 and Ch.24

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13: Philip J. Davis

… ►The surface color map can be changed from height-based to phase-based for complex valued functions, and density plots can be generated through strategic scaling. …

14: 7 Error Functions, Dawson’s and Fresnel Integrals

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15: 14.24 Analytic Continuation

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14.24.1 $$P_{\nu }^{-\mu }\left(z{\mathrm{e}}^{s\pi \mathrm{i}}\right)={\mathrm{e}}^{s\nu \pi \mathrm{i}}{P}_{\nu }^{-\mu }\left(z\right)+\frac{2\mathrm{i}\sin \left(\left(\nu +\frac{1}{2}\right)s\pi \right){\mathrm{e}}^{-s\pi \mathrm{i}/2}}{\cos \left(\nu \pi \right)\mathrm{\Gamma }\left(\mu -\nu \right)}{𝑸}_{\nu }^{\mu }\left(z\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $z$: complex variable, $\mu$: general order, $\nu$: general degree and $s$: arbitrary integer

Referenced by:

§14.24, §14.24

Permalink:

http://dlmf.nist.gov/14.24.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.24 and Ch.14

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14.24.2 $${𝑸}_{\nu }^{\mu }\left(z{\mathrm{e}}^{s\pi \mathrm{i}}\right)={(-1)}^s{\mathrm{e}}^{-s\nu \pi \mathrm{i}}{𝑸}_{\nu }^{\mu }\left(z\right),$$

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Symbols:

${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $\mu$: general order, $\nu$: general degree and $s$: arbitrary integer

Referenced by:

§14.23, §14.24

Permalink:

http://dlmf.nist.gov/14.24.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.24 and Ch.14

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14.24.3 $$P_{\nu ,s}^{-\mu }\left(z\right)={\mathrm{e}}^{s\mu \pi \mathrm{i}}{P}_{\nu }^{-\mu }\left(z\right),$$

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre 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, $z$: complex variable, $\mu$: general order, $\nu$: general degree and $s$: arbitrary integer

Permalink:

http://dlmf.nist.gov/14.24.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.24 and Ch.14

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14.24.4 $${𝑸}_{\nu ,s}^{\mu }\left(z\right)={\mathrm{e}}^{-s\mu \pi \mathrm{i}}{𝑸}_{\nu }^{\mu }\left(z\right)-\frac{\pi \mathrm{i}\sin \left(s\mu \pi \right)}{\sin \left(\mu \pi \right)\mathrm{\Gamma }\left(\nu -\mu +1\right)}{P}_{\nu }^{-\mu }\left(z\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $z$: complex variable, $\mu$: general order, $\nu$: general degree and $s$: arbitrary integer

Referenced by:

§14.24

Permalink:

http://dlmf.nist.gov/14.24.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.24 and Ch.14

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16: 14.23 Values on the Cut

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14.23.1 $$P_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)={\mathrm{e}}^{\mp \mu \pi \mathrm{i}/2}{𝖯}_{\nu }^{\mu }\left(x\right),$$

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Symbols:

: Ferrers function of the first kind, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre 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, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.23

Permalink:

http://dlmf.nist.gov/14.23.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.23 and Ch.14

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14.23.2 $${𝑸}_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)=\frac{{\mathrm{e}}^{\pm \mu \pi \mathrm{i}/2}}{\mathrm{\Gamma }\left(\nu +\mu +1\right)}\left({𝖰}_{\nu }^{\mu }\left(x\right)\mp \frac{1}{2}\pi \mathrm{i}{𝖯}_{\nu }^{\mu }\left(x\right)\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, : Ferrers function of the second kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.23

Permalink:

http://dlmf.nist.gov/14.23.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.23 and Ch.14

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14.23.4 $${𝖯}_{\nu }^{\mu }\left(x\right)={\mathrm{e}}^{\pm \mu \pi \mathrm{i}/2}{P}_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right),$$

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Symbols:

: Ferrers function of the first kind, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre 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, $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.3.2 (corrected)

Referenced by:

§14.23, §14.23, §14.3(iv)

Permalink:

http://dlmf.nist.gov/14.23.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.23 and Ch.14

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14.23.5 $${𝖰}_{\nu }^{\mu }\left(x\right)=\frac{1}{2}\mathrm{\Gamma }\left(\nu +\mu +1\right)\left({\mathrm{e}}^{-\mu \pi \mathrm{i}/2}{𝑸}_{\nu }^{\mu }\left(x+\mathrm{i}0\right)+{\mathrm{e}}^{\mu \pi \mathrm{i}/2}{𝑸}_{\nu }^{\mu }\left(x-\mathrm{i}0\right)\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.3.4 (modified)

Referenced by:

§14.23

Permalink:

http://dlmf.nist.gov/14.23.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.23 and Ch.14

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14.23.6 $${𝖰}_{\nu }^{\mu }\left(x\right)={\mathrm{e}}^{\mp \mu \pi \mathrm{i}/2}\mathrm{\Gamma }\left(\nu +\mu +1\right){𝑸}_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)\pm \frac{1}{2}\pi \mathrm{i}{\mathrm{e}}^{\pm \mu \pi \mathrm{i}/2}{P}_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.23, §14.23

Permalink:

http://dlmf.nist.gov/14.23.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §14.23 and Ch.14

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17: 14.15 Uniform Asymptotic Approximations

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14.15.4 $${𝖯}_{\nu }^{-\mu }\left(x\right)=\frac{1}{\mathrm{\Gamma }\left(\mu +1\right)}{\left(1-{\alpha }^2\right)}^{-\mu /2}{\left(\frac{1-\alpha }{1+\alpha }\right)}^{(\nu /2)+(1/4)}{\left(\frac{p}{x}\right)}^{1/2}{\mathrm{e}}^{-\mu \rho }\left(1+O\left(\frac{1}{\mu }\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $\mathrm{e}$: base of natural logarithm, $x$: real variable, $\mu$: general order, $\nu$: general degree, $\alpha$ , $p$ and $\rho$

Referenced by:

§14.15(ii)

Permalink:

http://dlmf.nist.gov/14.15.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(ii), §14.15 and Ch.14

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14.15.9 $${𝑸}_{\nu }^{\mu }\left(x\right)={\left(\frac{\pi }{2}\right)}^{1/2}{\left(\frac{\mathrm{e}}{\mu }\right)}^{\nu +(1/2)}{\left(\frac{1-\alpha }{1+\alpha }\right)}^{\mu /2}{\left(1-{\alpha }^2\right)}^{-(\nu /2)-(1/4)}{\left(\frac{{\alpha }^2+{\eta }^2}{{\alpha }^2\left(x^2-1\right)+1}\right)}^{1/4}{I}_{\nu +\frac{1}{2}}\left(\mu \eta \right)\left(1+O\left(\frac{1}{\mu }\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: 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, $x$: real variable, $\mu$: general order, $\nu$: general degree, $\alpha$ and $\eta$

Permalink:

http://dlmf.nist.gov/14.15.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(ii), §14.15 and Ch.14

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14.15.20 $$\beta ={\mathrm{e}}^{\mu }{\left(\frac{\nu -\mu +\frac{1}{2}}{\nu +\mu +\frac{1}{2}}\right)}^{(\nu /2)+(1/4)}{\left({\left(\nu +\frac{1}{2}\right)}^2-{\mu }^2\right)}^{-\mu /2},$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\mu$: general order, $\nu$: general degree and $\beta$

Permalink:

http://dlmf.nist.gov/14.15.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(iv), §14.15 and Ch.14

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18: 5.9 Integral Representations

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5.9.10 $$\mathrm{Ln}\mathrm{\Gamma }\left(z\right)=\left(z-\frac{1}{2}\right)\ln z-z+\frac{1}{2}\ln \left(2\pi \right)+2{\int }_0^{\mathrm{\infty }}\frac{\arctan \left(t/z\right)}{{\mathrm{e}}^{2\pi t}-1}dt,$$

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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, $\arctan z$: arctangent function, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

6.1.50

Referenced by:

(5.9.10_1), §5.9(i), §5.9(ii), Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8)

Permalink:

http://dlmf.nist.gov/5.9.E10

Encodings:

pMML, png, TeX

Addition (effective with 1.0.10):

To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\mathrm{Ln}\mathrm{\Gamma }\left(z\right)$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }\left(z\right)$ was used, where $\ln$ is the principal branch of the logarithm.

Suggested 2015-02-13 by Philippe Spindel

See also:

Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

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5.9.10_1 $$\mathrm{Ln}\mathrm{\Gamma }\left(z\right)=\left(z-\frac{1}{2}\right)\ln z-z+\frac{1}{2}\ln \left(2\pi \right)-\frac{z}{\pi }{\int }_0^{\mathrm{\infty }}\frac{\ln \left(1-{\mathrm{e}}^{-2\pi t}\right)}{t^2+z^2}dt,$$

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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, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function and $z$: complex variable

Proof sketch:

Can be obtained from (5.9.10) via integration by parts.

Referenced by:

§5.9(i), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/5.9.E10_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

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5.9.10_2 $$\mathrm{Ln}\mathrm{\Gamma }\left(z\right)=\left(z-\frac{1}{2}\right)\ln z-z+\frac{1}{2}\ln \left(2\pi \right)+{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}\left(\frac{1}{{\mathrm{e}}^t-1}-\frac{1}{t}+\frac{1}{2}\right)\frac{dt}{t},$$

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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, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function and $z$: complex variable

Source:

Whittaker and Watson (1927, §12.31)

Referenced by:

§5.9(i), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/5.9.E10_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

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19: 2.11 Remainder Terms; Stokes Phenomenon

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2.11.5 $$E_p\left(z\right)=\frac{{\mathrm{e}}^{-z}{z}^{p-1}}{\mathrm{\Gamma }\left(p\right)}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-zt}{t}^{p-1}}{1+t}dt$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral and $\int$: integral

Referenced by:

§2.11(iii)

Permalink:

http://dlmf.nist.gov/2.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(ii), §2.11 and Ch.2

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2.11.6 $$E_p\left(z\right)\sim \frac{{\mathrm{e}}^{-z}}{z}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(p\right)}_s}{z^s}$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm and $E_p\left(z\right)$: generalized exponential integral

Referenced by:

§2.11(ii), §2.11(iii), §2.11(iii), §2.11(iv), §2.11(iv)

Permalink:

http://dlmf.nist.gov/2.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(ii), §2.11 and Ch.2

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2.11.7 $$E_p\left(z\right)\sim \frac{2\pi \mathrm{i}{\mathrm{e}}^{-p\pi \mathrm{i}}}{\mathrm{\Gamma }\left(p\right)}{z}^{p-1}+\frac{{\mathrm{e}}^{-z}}{z}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(p\right)}_s}{z^s},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral and $\mathrm{i}$: imaginary unit

Referenced by:

§2.11(ii), §2.11(iv), §2.11(iv), §2.11(iv)

Permalink:

http://dlmf.nist.gov/2.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(ii), §2.11 and Ch.2

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2.11.10 $$E_p\left(z\right)=\frac{{\mathrm{e}}^{-z}}{z}\sum _{s=0}^{n-1}{(-1)}^s\frac{{\left(p\right)}_s}{z^s}+{(-1)}^n\frac{2\pi }{\mathrm{\Gamma }\left(p\right)}{z}^{p-1}{F}_{n+p}\left(z\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral and $F_p\left(z\right)$: terminant function

Referenced by:

§2.11(iii), §2.11(iv)

Permalink:

http://dlmf.nist.gov/2.11.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(iii), §2.11 and Ch.2

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2.11.11 $$F_{n+p}\left(z\right)=\frac{{\mathrm{e}}^{-z}}{2\pi }{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-zt}{t}^{n+p-1}}{1+t}dt=\frac{\mathrm{\Gamma }\left(n+p\right)}{2\pi }\frac{E_{n+p}\left(z\right)}{z^{n+p-1}}.$$

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Defines:

$F_p\left(z\right)$: terminant function

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, $E_p\left(z\right)$: generalized exponential integral and $\int$: integral

Permalink:

http://dlmf.nist.gov/2.11.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(iii), §2.11 and Ch.2

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20: 19 Elliptic Integrals

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