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11: 28.26 Asymptotic Approximations for Large $q$

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28.26.1 $${\mathrm{Mc}}_m^{(3)}(z,h)=\frac{{\mathrm{e}}^{\mathrm{i}\varphi }}{{(\pi h\cosh z)}^{1/2}}\left({\mathrm{Fc}}_m(z,h)-\mathrm{i}{\mathrm{Gc}}_m(z,h)\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable and $\varphi$

A&S Ref:

20.9.7 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.26.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

►

28.26.2 $$\mathrm{i}{\mathrm{Ms}}_{m+1}^{(3)}(z,h)=\frac{{\mathrm{e}}^{\mathrm{i}\varphi }}{{(\pi h\cosh z)}^{1/2}}\left({\mathrm{Fs}}_m(z,h)-\mathrm{i}{\mathrm{Gs}}_m(z,h)\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable and $\varphi$

A&S Ref:

20.9.7 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.26.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

… ►

28.26.3 $$\varphi =2h\sinh z-\left(m+\frac{1}{2}\right)\arctan \left(\sinh z\right).$$

ⓘ

Symbols:

$\sinh z$: hyperbolic sine function, $\arctan z$: arctangent function, $m$: integer, $h$: parameter, $z$: complex variable and $\varphi$

A&S Ref:

20.9.8 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.26.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

… ►

28.26.4 $${\mathrm{Fc}}_m(z,h)\sim 1+\frac{s}{8h{\cosh }^{2}z}+\frac{1}{2^{11}{h}^2}\left(\frac{s^4+86s^2+105}{{\cosh }^{4}z}-\frac{s^4+22s^2+57}{{\cosh }^{2}z}\right)+\frac{1}{2^{14}{h}^3}\left(-\frac{s^5+14s^3+33s}{{\cosh }^{2}z}-\frac{2s^5+124s^3+1122s}{{\cosh }^{4}z}+\frac{3s^5+290s^3+1627s}{{\cosh }^{6}z}\right)+\mathrm{\cdots },$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\cosh z$: hyperbolic cosine function, $m$: integer, $h$: parameter and $z$: complex variable

A&S Ref:

20.9.9 (in slightly different notation)

Referenced by:

§28.26(i)

Permalink:

http://dlmf.nist.gov/28.26.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

►

28.26.5 $${\mathrm{Gc}}_m(z,h)\sim \frac{\sinh z}{{\cosh }^{2}z}\left(\frac{s^2+3}{2^{5}h}+\frac{1}{2^9{h}^2}\left(s^3+3s+\frac{4s^3+44s}{{\cosh }^{2}z}\right)+\frac{1}{2^{14}{h}^3}\left(5s^4+34s^2+9-\frac{s^6-47s^4+667s^2+2835}{12{\cosh }^{2}z}+\frac{s^6+505s^4+12139s^2+10395}{12{\cosh }^{4}z}\right)\right)+\mathrm{\cdots }.$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $m$: integer, $h$: parameter and $z$: complex variable

A&S Ref:

20.9.10 (in slightly different notation)

Referenced by:

§28.26(i)

Permalink:

http://dlmf.nist.gov/28.26.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

…

12: 29.7 Asymptotic Expansions

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29.7.3 $${\tau }_0=\frac{1}{2^3}(1+k^2)(1+p^2),$$

ⓘ

Symbols:

$p$: nonnegative integer, $k$: real parameter and ${\tau }_j$: coefficients

Permalink:

http://dlmf.nist.gov/29.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

►

29.7.4 $${\tau }_1=\frac{p}{2^6}({(1+k^2)}^2(p^2+3)-4k^2(p^2+5)).$$

ⓘ

Symbols:

$p$: nonnegative integer, $k$: real parameter and ${\tau }_j$: coefficients

Permalink:

http://dlmf.nist.gov/29.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

… ►

29.7.6 $${\tau }_2=\frac{1}{2^{10}}(1+k^2){(1-k^2)}^2(5p^4+34p^2+9),$$

ⓘ

Symbols:

$p$: nonnegative integer, $k$: real parameter and ${\tau }_j$: coefficients

Permalink:

http://dlmf.nist.gov/29.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

►

29.7.7 $${\tau }_3=\frac{p}{2^{14}}({(1+k^2)}^4(33p^4+410p^2+405)-24k^2{(1+k^2)}^2(7p^4+90p^2+95)+16k^4(9p^4+130p^2+173)),$$

ⓘ

Symbols:

$p$: nonnegative integer, $k$: real parameter and ${\tau }_j$: coefficients

Permalink:

http://dlmf.nist.gov/29.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

►

29.7.8 $${\tau }_4=\frac{1}{2^{16}}({(1+k^2)}^5(63p^6+1260p^4+2943p^2+486)-8k^2{(1+k^2)}^3(49p^6+1010p^4+2493p^2+432)+16k^4(1+k^2)(35p^6+760p^4+2043p^2+378)).$$

ⓘ

Symbols:

$p$: nonnegative integer, $k$: real parameter and ${\tau }_j$: coefficients

Permalink:

http://dlmf.nist.gov/29.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §29.7(i), §29.7 and Ch.29

…

13: 28.10 Integral Equations

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28.10.1 $$\frac{2}{\pi }{\int }_0^{\pi /2}\cos \left(2h\cos z\cos t\right){\mathrm{ce}}_{2n}(t,h^2)dt=\frac{A_0^{2n}(h^2)}{{\mathrm{ce}}_{2n}(\frac{1}{2}\pi ,h^2)}{\mathrm{ce}}_{2n}(z,h^2),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential, $\int$: integral, $h$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

A&S Ref:

20.7.20 (in different form)

Permalink:

http://dlmf.nist.gov/28.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.10(i), §28.10 and Ch.28

►

28.10.2 $$\frac{2}{\pi }{\int }_0^{\pi /2}\cosh \left(2h\sin z\sin t\right){\mathrm{ce}}_{2n}(t,h^2)dt=\frac{A_0^{2n}(h^2)}{{\mathrm{ce}}_{2n}(0,h^2)}{\mathrm{ce}}_{2n}(z,h^2),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential, $\cosh z$: hyperbolic cosine function, $\int$: integral, $\sin z$: sine function, $h$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

A&S Ref:

20.7.21 (in different form)

Permalink:

http://dlmf.nist.gov/28.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.10(i), §28.10 and Ch.28

►

28.10.3 $$\frac{2}{\pi }{\int }_0^{\pi /2}\sin \left(2h\cos z\cos t\right){\mathrm{ce}}_{2n+1}(t,h^2)dt=-\frac{h{A}_1^{2n+1}(h^2)}{{\mathrm{ce}}_{2n+1}^{\prime }(\frac{1}{2}\pi ,h^2)}{\mathrm{ce}}_{2n+1}(z,h^2),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential, $\int$: integral, $\sin z$: sine function, $h$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

A&S Ref:

20.7.20 (in different form)

Permalink:

http://dlmf.nist.gov/28.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.10(i), §28.10 and Ch.28

►

28.10.4 $$\frac{2}{\pi }{\int }_0^{\pi /2}\cos z\cos t\cosh \left(2h\sin z\sin t\right){\mathrm{ce}}_{2n+1}(t,h^2)dt=\frac{A_1^{2n+1}(h^2)}{2{\mathrm{ce}}_{2n+1}(0,h^2)}{\mathrm{ce}}_{2n+1}(z,h^2),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential, $\cosh z$: hyperbolic cosine function, $\int$: integral, $\sin z$: sine function, $h$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

A&S Ref:

20.7.21 (in different form)

Permalink:

http://dlmf.nist.gov/28.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.10(i), §28.10 and Ch.28

►

28.10.5 $$\frac{2}{\pi }{\int }_0^{\pi /2}\sinh \left(2h\sin z\sin t\right){\mathrm{se}}_{2n+1}(t,h^2)dt=\frac{h{B}_1^{2n+1}(h^2)}{{\mathrm{se}}_{2n+1}^{\prime }(0,h^2)}{\mathrm{se}}_{2n+1}(z,h^2),$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential, $\sinh z$: hyperbolic sine function, $\int$: integral, $\sin z$: sine function, $h$: parameter, $n$: integer, $z$: complex variable and $B_m(q)$: Fourier coefficient

A&S Ref:

20.7.23 (in different form)

Permalink:

http://dlmf.nist.gov/28.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.10(i), §28.10 and Ch.28

…

14: 28.2 Definitions and Basic Properties

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28.2.19 $$q{c}_{2n+2}-\left(a-{(\nu +2n)}^2\right)c_{2n}+q{c}_{2n-2}=0,$$ $n\in \mathbb{Z}$.

ⓘ

Symbols:

$\in$: element of, $\mathbb{Z}$: set of all integers, $q=h^2$: parameter, $n$: integer, $\nu$: complex parameter, $a$: parameter and $c_{2n}$: coefficients

A&S Ref:

20.3.12 20.3.13

Referenced by:

§28.2(iv)

Permalink:

http://dlmf.nist.gov/28.2.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(iv), §28.2 and Ch.28

… ►

28.2.21

ⓘ

Symbols:

$a_n\left(q\right)$: eigenvalues of Mathieu equation, $\pi$: the ratio of the circumference of a circle to its diameter, $q=h^2$: parameter, $n$: integer, $a$: parameter, $w(z)$: Mathieu’s equation solution and $w_{\text{I}}(z;a,q)$: fundamental solution

Referenced by:

§28.2(vi), §28.29(ii), §28.7

Permalink:

http://dlmf.nist.gov/28.2.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(v), §28.2 and Ch.28

… ►

28.2.22

ⓘ

Symbols:

$b_n\left(q\right)$: eigenvalues of Mathieu equation, $\pi$: the ratio of the circumference of a circle to its diameter, $q=h^2$: parameter, $n$: integer, $a$: parameter, $w(z)$: Mathieu’s equation solution and $w_{\text{II}}(z;a,q)$: fundamental solution

Referenced by:

§28.2(vi), §28.29(ii), §28.7

Permalink:

http://dlmf.nist.gov/28.2.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(v), §28.2 and Ch.28

… ►

28.2.31 $${\int }_0^{2\pi }{\mathrm{ce}}_m(x,q){\mathrm{ce}}_n(x,q)dx=0,$$ $n\ne m$,

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential, $\int$: integral, $m$: integer, $q=h^2$: parameter, $n$: integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/28.2.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(vi), §28.2 and Ch.28

►

28.2.32 $${\int }_0^{2\pi }{\mathrm{se}}_m(x,q){\mathrm{se}}_n(x,q)dx=0,$$ $n\ne m$,

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential, $\int$: integral, $m$: integer, $q=h^2$: parameter, $n$: integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/28.2.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(vi), §28.2 and Ch.28

…

15: 31.15 Stieltjes Polynomials

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31.15.2 $$\sum _{j=1}^N\frac{{\gamma }_j/2}{z_k-a_j}+\sum _{\begin{array}{c}\hfill j=1\hfill \\ \hfill j\ne k\hfill \end{array}}^n\frac{1}{z_k-z_j}=0,$$ $k=1,2,\mathrm{\dots },n$.

ⓘ

Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter and $N+1$: number of singularities

Referenced by:

§31.15(ii)

Permalink:

http://dlmf.nist.gov/31.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(ii), §31.15 and Ch.31

… ►

31.15.3 $$\sum _{j=1}^N\frac{{\gamma }_j}{t_k-a_j}+\sum _{j=1}^{n-1}\frac{1}{t_k-z_j^{\prime }}=0.$$

ⓘ

Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $N+1$: number of singularities and $t_k$: zero of $V(z)$

Permalink:

http://dlmf.nist.gov/31.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(ii), §31.15 and Ch.31

… ►

31.15.6 $j=1,2,\mathrm{\dots },N-1$,

ⓘ

Symbols:

$j$: nonnegative integer, $a$: complex parameter and $N+1$: number of singularities

Referenced by:

§31.15(iii)

Permalink:

http://dlmf.nist.gov/31.15.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(ii), §31.15 and Ch.31

… ►

31.15.7 $$q_j={\gamma }_j\sum _{k=1}^n\frac{1}{z_k-a_j},$$ $j=1,2,\mathrm{\dots },N$.

ⓘ

Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $N+1$: number of singularities

Permalink:

http://dlmf.nist.gov/31.15.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(ii), §31.15 and Ch.31

… ►If the exponent and singularity parameters satisfy (31.15.5)–(31.15.6), then for every multi-index $\mathbf{m}=(m_1,m_2,\mathrm{\dots },m_{N-1})$, where each $m_j$ is a nonnegative integer, there is a unique Stieltjes polynomial with $m_j$ zeros in the open interval $(a_j,a_{j+1})$ for each $j=1,2,\mathrm{\dots },N-1$. …

16: 28.8 Asymptotic Expansions for Large $q$

… ►

28.8.4 $$U_m(\xi )\sim D_m\left(\xi \right)-\frac{1}{2^{6}h}\left(D_{m+4}\left(\xi \right)-4!\left(\genfrac{}{}{0pt}{}{m}{4}\right)D_{m-4}\left(\xi \right)\right)+\frac{1}{2^{13}{h}^2}\left(D_{m+8}\left(\xi \right)-2^5(m+2)D_{m+4}\left(\xi \right)+4!{\mathrm{\hspace{0.17em}2}}^5(m-1)\left(\genfrac{}{}{0pt}{}{m}{4}\right)D_{m-4}\left(\xi \right)+8!\left(\genfrac{}{}{0.0pt}{}{m}{8}\right)D_{m-8}\left(\xi \right)\right)+\mathrm{\cdots },$$

ⓘ

Defines:

$U_m(\xi )$: function (locally)

Symbols:

$D_{\nu }\left(z\right)$: parabolic cylinder function, $\sim$: Poincaré asymptotic expansion, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), $m$: integer, $h$: parameter and $\xi$: variable

A&S Ref:

20.9.17 (in different form)

Permalink:

http://dlmf.nist.gov/28.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(ii), §28.8 and Ch.28

►

28.8.5 $$V_m(\xi )\sim \frac{1}{2^{4}h}\left(-D_{m+2}\left(\xi \right)-m(m-1)D_{m-2}\left(\xi \right)\right)+\frac{1}{2^{10}{h}^2}\left(D_{m+6}\left(\xi \right)+(m^2-25m-36)D_{m+2}\left(\xi \right)-m(m-1)(m^2+27m-10)D_{m-2}\left(\xi \right)-6!\left(\genfrac{}{}{0.0pt}{}{m}{6}\right)D_{m-6}\left(\xi \right)\right)+\mathrm{\cdots },$$

ⓘ

Defines:

$V_m(\xi )$: function (locally)

Symbols:

$D_{\nu }\left(z\right)$: parabolic cylinder function, $\sim$: Poincaré asymptotic expansion, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), $m$: integer, $h$: parameter and $\xi$: variable

A&S Ref:

20.9.18 (in slightly different form)

Referenced by:

Erratum (V1.0.16) for Equation (28.8.5)

Permalink:

http://dlmf.nist.gov/28.8.E5

Encodings:

pMML, png, TeX

Error (effective with 1.0.16):

Originally the $-$ in front of the $6!$ was given incorrectly as $+$.

Suggested 2017-02-02 by Daniel Karlsson

See also:

Annotations for §28.8(ii), §28.8 and Ch.28

… ►

28.8.9 $$W_m^{\pm }(x)=\frac{{\mathrm{e}}^{\pm 2h\sin x}}{{(\cos x)}^{m+1}}\{\begin{array}{cc}{\left(\cos \left(\frac{1}{2}x+\frac{1}{4}\pi \right)\right)}^{2m+1},\hfill & \\ {\left(\sin \left(\frac{1}{2}x+\frac{1}{4}\pi \right)\right)}^{2m+1},\hfill & \end{array}$$

ⓘ

Defines:

$W_m^{\pm }$: function (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\sin z$: sine function, $m$: integer, $h$: parameter and $x$: real variable

A&S Ref:

20.9.13 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(iii), §28.8 and Ch.28

… ►

28.8.11 $$P_m(x)\sim 1+\frac{s}{2^{3}h{\cos }^{2}x}+\frac{1}{h^2}\left(\frac{s^4+86s^2+105}{2^{11}{\cos }^{4}x}-\frac{s^4+22s^2+57}{2^{11}{\cos }^{2}x}\right)+\mathrm{\cdots },$$

ⓘ

Defines:

$P_m(x)$ (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion, $\cos z$: cosine function, $m$: integer, $h$: parameter and $x$: real variable

A&S Ref:

20.9.14 (in different form)

Permalink:

http://dlmf.nist.gov/28.8.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(iii), §28.8 and Ch.28

►

28.8.12 $$Q_m(x)\sim \frac{\sin x}{{\cos }^{2}x}\left(\frac{1}{2^{5}h}(s^2+3)+\frac{1}{2^9{h}^2}\left(s^3+3s+\frac{4s^3+44s}{{\cos }^{2}x}\right)\right)+\mathrm{\cdots }.$$

ⓘ

Defines:

$Q_m(x)$ (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion, $\cos z$: cosine function, $\sin z$: sine function, $m$: integer, $h$: parameter and $x$: real variable

A&S Ref:

20.9.14 (in different form)

Permalink:

http://dlmf.nist.gov/28.8.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(iii), §28.8 and Ch.28

…

17: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

… ►

31.5.1

ⓘ

Symbols:

$\gamma$: real or complex parameter, $n$: nonnegative integer, $a$: complex parameter, $P_j$: coefficient, $Q_j$: coefficient and $R_j$: coefficient

Permalink:

http://dlmf.nist.gov/31.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.5 and Ch.31

… ►

31.5.2 $${\mathit{Hp}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)=H\mathrm{\ell }(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$$

ⓘ

Defines:

${\mathit{Hp}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$: Heun polynomials

Symbols:

$H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $m$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.5 and Ch.31

…

18: 24.17 Mathematical Applications

… ►Let $0\le h\le 1$ and $a,m$, and $n$ be integers such that $n>a$, $m>0$, and $f^{(m)}(x)$ is absolutely integrable over $[a,n]$. … ►

24.17.1 $$\sum _{j=a}^{n-1}{(-1)}^{j}f(j+h)=\frac{1}{2}\sum _{k=0}^{m-1}\frac{E_k\left(h\right)}{k!}\left({(-1)}^{n-1}{f}^{(k)}(n)+{(-1)}^a{f}^{(k)}(a)\right)+R_m(n),$$

ⓘ

Symbols:

$E_n\left(x\right)$: Euler polynomials, $!$: factorial (as in $n!$), $j$: integer, $k$: integer, $m$: integer, $h$: parameter, $a$: integer, $n$: integer, $f^{(m)}(x)$: integrable function and $R_m(n)$: remainder

Permalink:

http://dlmf.nist.gov/24.17.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §24.17(i), §24.17(i), §24.17 and Ch.24

… ►

24.17.2 $$R_m(n)=\frac{1}{2(m-1)!}{\int }_a^n{f}^{(m)}(x){\tilde{E}}_{m-1}\left(h-x\right)dx.$$

ⓘ

Symbols:

$dx$: differential, $!$: factorial (as in $n!$), $\int$: integral, ${\tilde{E}}_n\left(x\right)$: periodic Euler functions, $m$: integer, $x$: real or complex, $h$: parameter, $a$: integer, $n$: integer, $f^{(m)}(x)$: integrable function and $R_m(n)$: remainder

Permalink:

http://dlmf.nist.gov/24.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §24.17(i), §24.17(i), §24.17 and Ch.24

…

19: 8.18 Asymptotic Expansions of $I_x(a,b)$

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8.18.1 $$I_x(a,b)=\mathrm{\Gamma }\left(a+b\right)x^a{(1-x)}^{b-1}\left(\sum _{k=0}^{n-1}\frac{1}{\mathrm{\Gamma }\left(a+k+1\right)\mathrm{\Gamma }\left(b-k\right)}{\left(\frac{x}{1-x}\right)}^k+O\left(\frac{1}{\mathrm{\Gamma }\left(a+n+1\right)}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $I_x(a,b)$: incomplete beta function, $k$: nonnegative integer, $n$: nonnegative integer, $a$: parameter, $b$: parameter and $x$: variable

Referenced by:

§8.18(i)

Permalink:

http://dlmf.nist.gov/8.18.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.18(i), §8.18 and Ch.8

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8.18.3 $$I_x(a,b)=\frac{\mathrm{\Gamma }\left(a+b\right)}{\mathrm{\Gamma }\left(a\right)}\left(\sum _{k=0}^{n-1}{d}_k{F}_k+O\left(a^{-n}\right)F_0\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $I_x(a,b)$: incomplete beta function, $\sim$: Poincaré asymptotic expansion, $k$: nonnegative integer, $n$: nonnegative integer, $a$: parameter, $b$: parameter, $x$: variable, $F_k$: functions and $d_k$: coefficients

Referenced by:

Erratum (V1.0.14) for Equation (8.18.3)

Permalink:

http://dlmf.nist.gov/8.18.E3

Encodings:

pMML, png, TeX

Addition (effective with 1.0.14):

Previously this equation appeared without the order estimate as $I_x(a,b)\sim \frac{\mathrm{\Gamma }\left(a+b\right)}{\mathrm{\Gamma }\left(a\right)}{\sum }_{k=0}^{\mathrm{\infty }}{d}_k{F}_k$. The range of $x$ was extended to include $1$.

See also:

Annotations for §8.18(ii), §8.18(ii), §8.18 and Ch.8

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8.18.4 $$a{F}_{k+1}=(k+b-a\xi )F_k+k\xi F_{k-1},$$

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Symbols:

$k$: nonnegative integer, $a$: parameter, $b$: parameter, $\xi$: change of variable and $F_k$: functions

Permalink:

http://dlmf.nist.gov/8.18.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §8.18(ii), §8.18(ii), §8.18 and Ch.8

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8.18.6 $${\left(\frac{1-{\mathrm{e}}^{-t}}{t}\right)}^{b-1}=\sum _{k=0}^{\mathrm{\infty }}{d}_k{(t-\xi )}^k.$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $k$: nonnegative integer, $b$: parameter, $\xi$: change of variable and $d_k$: coefficients

Permalink:

http://dlmf.nist.gov/8.18.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §8.18(ii), §8.18(ii), §8.18 and Ch.8

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8.18.9 $$I_x(a,b)\sim \frac{1}{2}\mathrm{erfc}\left(-\eta \sqrt{b/2}\right)+\frac{1}{\sqrt{2\pi (a+b)}}{\left(\frac{x}{x_0}\right)}^a{\left(\frac{1-x}{1-x_0}\right)}^b\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{c}_k(\eta )}{{(a+b)}^k},$$

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Symbols:

$I_x(a,b)$: incomplete beta function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $k$: nonnegative integer, $a$: parameter, $b$: parameter, $x$: variable and $c_k(\eta )$: coefficients

Permalink:

http://dlmf.nist.gov/8.18.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §8.18(ii), §8.18(ii), §8.18 and Ch.8

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20: 28.25 Asymptotic Expansions for Large $\mathrm{\Re }z$

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28.25.1 $${\mathrm{M}}_{\nu }^{(3,4)}(z,h)\sim \frac{{\mathrm{e}}^{\pm \mathrm{i}\left(2h\cosh z-\left(\frac{1}{2}\nu +\frac{1}{4}\right)\pi \right)}}{{\left(\pi h(\cosh z+1)\right)}^{\frac{1}{2}}}\sum _{m=0}^{\mathrm{\infty }}\frac{D_m^{\pm }}{{\left(\mp 4\mathrm{i}h(\cosh z+1)\right)}^m},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable, $\nu$: complex parameter and $D_m^{\pm }$: coefficients

A&S Ref:

20.9.1 (for 3 and for integer $\nu$) 20.9.3 (for 4 and for integer $\nu$)

Referenced by:

§28.25

Permalink:

http://dlmf.nist.gov/28.25.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.25 and Ch.28

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28.25.3 $$(m+1)D_{m+1}^{\pm }+\left({(m+\frac{1}{2})}^2\pm (m+\frac{1}{4})8\mathrm{i}h+2h^2-a\right)D_m^{\pm }\pm (m-\frac{1}{2})\left(8\mathrm{i}hm\right)D_{m-1}^{\pm }=0,$$ $m\ge 0$.

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Symbols:

$\mathrm{i}$: imaginary unit, $m$: integer, $h$: parameter, $a$: parameter and $D_m^{\pm }$: coefficients

A&S Ref:

20.9.2 (for $+$) 20.9.4 (for $-$)

Permalink:

http://dlmf.nist.gov/28.25.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.25 and Ch.28

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