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1: Need Help?

… ►We have also tried to use the best technologies available in order to make this information useful and accessible. … ►

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2: About MathML

… ►The DLMF uses MathML Core, a subset designed specifically for browsers. … ►However, the MathJax javascript library can also used to render the MathML, albeit more slowly. … ►Of course you are encouraged to use a modern, up-to-date browser. … ►DLMF uses the STIX web font to provide a consistent coverage. ►For other browsers, you may see a ? or a box like indicating missing symbols, and thus insufficient fonts. …

3: 18.36 Miscellaneous Polynomials

… ►This lays the foundation for consideration of exceptional OP’s wherein a finite number of (possibly non-sequential) polynomial orders are missing, from what is a complete set even in their absence. … ►Exceptional type I $X_m$-EOP’s, form a complete orthonormal set with respect to a positive measure, but the lowest order polynomial in the set is of order $m$, or, said another way, the first $m$ polynomial orders, $0,1,\mathrm{\dots },m-1$ are missing. The exceptional type III $X_m$-EOP’s are missing orders $1,\mathrm{\dots },m$. …EOP’s are non-classical in that not only are certain polynomial orders missing, but, also, not all EOP polynomial zeros are within the integration range of their generating measure, and EOP-orthogonality properties do not allow development of Gaussian-type quadratures. … ►The type III $X_2$-Hermite EOP’s, missing polynomial orders $1$ and $2$, are the complete set of polynomials, with real coefficients and defined explicitly as …

4: 28.1 Special Notation

… ► ►►

${\mathrm{ce}}_{\nu }(z,q)$,

${\mathrm{se}}_{\nu }(z,q)$,

${\mathrm{fe}}_n(z,q)$,

${\mathrm{ge}}_n(z,q)$,

${\mathrm{me}}_{\nu }(z,q)$,

… ► ►►►

${\mathrm{Ce}}_{\nu }(z,q)$,	${\mathrm{Se}}_{\nu }(z,q)$,	${\mathrm{Fe}}_n(z,q)$,	${\mathrm{Ge}}_n(z,q)$,
${\mathrm{Me}}_{\nu }(z,q)$,	${\mathrm{M}}_{\nu }^{(j)}(z,h)$,	${\mathrm{Mc}}_n^{(j)}(z,h)$,	${\mathrm{Ms}}_n^{(j)}(z,h)$,
…

… ►

$${\mathrm{Me}}_n^{(1,2)}(z,q)=\sqrt{\frac{1}{2}\pi }{g}_{e,n}(h){\mathrm{ce}}_n(0,q){\mathrm{Mc}}_n^{(3,4)}(z,h),$$

… ►Arscott (1964b) also uses $-\mathrm{i}\mu$ for $\nu$. … ►

$$F_{\nu }(z)={\mathrm{Me}}_{\nu }(z,q).$$

…

5: 28.19 Expansions in Series of ${\mathrm{me}}_{\nu +2n}$ Functions

§28.19 Expansions in Series of ${\mathrm{me}}_{\nu +2n}$ Functions

… ►

28.19.2 $$f(z)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{f}_n{\mathrm{me}}_{\nu +2n}(z,q),$$

ⓘ

Defines:

$f(z)$: function (locally)

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $q=h^2$: parameter, $n$: integer, $z$: complex variable, $\nu$: complex parameter and $f_n$: coefficients

Referenced by:

§28.19

Permalink:

http://dlmf.nist.gov/28.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.19 and Ch.28

… ►

28.19.3 $$f_n=\frac{1}{\pi }{\int }_0^{\pi }f(z){\mathrm{me}}_{\nu +2n}(-z,q)dz.$$

ⓘ

Defines:

$f_n$: coefficients (locally)

Symbols:

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

Permalink:

http://dlmf.nist.gov/28.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.19 and Ch.28

… ►

28.19.4 $${\mathrm{e}}^{\mathrm{i}\nu z}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_{-2n}^{\nu +2n}(q){\mathrm{me}}_{\nu +2n}(z,q),$$

ⓘ

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $q=h^2$: parameter, $n$: integer, $z$: complex variable, $\nu$: complex parameter and $c_{2m}(q)$: coefficients

Permalink:

http://dlmf.nist.gov/28.19.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.19, §28.19 and Ch.28

…

6: 28.28 Integrals, Integral Representations, and Integral Equations

… ►

28.28.7 $$\frac{1}{\pi }{\int }_{{ℒ}_j}{\mathrm{e}}^{2\mathrm{i}hw}{\mathrm{me}}_{\nu }(t,h^2)dt={\mathrm{e}}^{\mathrm{i}\nu \pi /2}{\mathrm{me}}_{\nu }(\alpha ,h^2){\mathrm{M}}_{\nu }^{(j)}(z,h),$$ $j=3,4$,

ⓘ

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu 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, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $j$: integer, $z$: complex variable, $\nu$: complex parameter, $w(z)$: Mathieu’s equation solution, $\alpha$: parameter and ${ℒ}_j$: integration paths

A&S Ref:

20.7.18 (in slightly different notation)

Referenced by:

§28.28(i)

Permalink:

http://dlmf.nist.gov/28.28.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(i), §28.28 and Ch.28

… ►With the parameter $h$ suppressed we use the notation … ►

28.28.27 $${\alpha }_{\nu ,m}^{(0)}=\frac{1}{2\pi }{\int }_0^{2\pi }\cos t{\mathrm{me}}_{\nu }(t,h^2){\mathrm{me}}_{-\nu -2m-1}(t,h^2)dt={(-1)}^m\frac{2\mathrm{i}}{\pi }\frac{{\mathrm{me}}_{\nu }(0,h^2){\mathrm{me}}_{-\nu -2m-1}(0,h^2)}{h{\mathrm{D}}_0(\nu ,\nu +2m+1,0)},$$

ⓘ

Defines:

${\alpha }_{\nu ,m}^{(s)}$ (locally)

Symbols:

${\mathrm{D}}_j(\nu ,\mu ,z)$: cross-products of modified Mathieu functions and their derivatives, ${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $m$: integer, $h$: parameter and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.28.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(iii), §28.28 and Ch.28

►

28.28.28 $${\alpha }_{\nu ,m}^{(1)}=\frac{1}{2\pi }{\int }_0^{2\pi }\sin t{\mathrm{me}}_{\nu }(t,h^2){\mathrm{me}}_{-\nu -2m-1}(t,h^2)dt={(-1)}^{m+1}\frac{2\mathrm{i}}{\pi }\frac{{\mathrm{me}}_{\nu }^{\prime }(0,h^2){\mathrm{me}}_{-\nu -2m-1}(0,h^2)}{h{\mathrm{D}}_1(\nu ,\nu +2m+1,0)}.$$

ⓘ

Symbols:

${\mathrm{D}}_j(\nu ,\mu ,z)$: cross-products of modified Mathieu functions and their derivatives, ${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $m$: integer, $h$: parameter, $\nu$: complex parameter and ${\alpha }_{\nu ,m}^{(s)}$

Permalink:

http://dlmf.nist.gov/28.28.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(iii), §28.28 and Ch.28

… ►

28.28.33 $${\gamma }_{\nu ,m}=\frac{1}{2\pi }{\int }_0^{2\pi }{\mathrm{me}}_{\nu }^{\prime }\left(t\right){\mathrm{me}}_{-\nu -2m}\left(t\right)dt={(-1)}^m\frac{4\mathrm{i}}{\pi }\frac{{\mathrm{me}}_{\nu }^{\prime }\left(0\right){\mathrm{me}}_{-\nu -2m}\left(0\right)}{{\mathrm{D}}_1(\nu ,\nu +2m,0)}.$$

ⓘ

Defines:

${\gamma }_{\nu ,m}$ (locally)

Symbols:

${\mathrm{D}}_j(\nu ,\mu ,z)$: cross-products of modified Mathieu functions and their derivatives, ${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $m$: integer, $h$: parameter and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.28.E33

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(iii), §28.28 and Ch.28

…

7: 28.12 Definitions and Basic Properties

… ►

§28.12(ii) Eigenfunctions ${\mathrm{me}}_{\nu }(z,q)$

… ►The Floquet solution with respect to $\nu$ is denoted by ${\mathrm{me}}_{\nu }(z,q)$. For $q=0$, … ►To complete the definitions of the ${\mathrm{me}}_{\nu }$ functions we set … ►These functions are real-valued for real $\nu$, real $q$, and $z=x$, whereas ${\mathrm{me}}_{\nu }(x,q)$ is complex. …

8: 28.23 Expansions in Series of Bessel Functions

… ►We use the following notations: … ►

28.23.2 $${\mathrm{me}}_{\nu }(0,h^2){\mathrm{M}}_{\nu }^{(j)}(z,h)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{(-1)}^n{c}_{2n}^{\nu }(h^2){𝒞}_{\nu +2n}^{(j)}(2h\cosh z),$$

ⓘ

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $n$: integer, $j$: integer, $z$: complex variable, $\nu$: complex parameter, $c_{2m}(q)$: coefficients and ${𝒞}_{\mu }^{(j)}$: cylinder functions

Permalink:

http://dlmf.nist.gov/28.23.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

►

28.23.3 $${\mathrm{me}}_{\nu }^{\prime }(0,h^2){\mathrm{M}}_{\nu }^{(j)}(z,h)=\mathrm{i}\tanh z\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{(-1)}^n(\nu +2n)c_{2n}^{\nu }(h^2){𝒞}_{\nu +2n}^{(j)}(2h\cosh z),$$

ⓘ

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $\mathrm{i}$: imaginary unit, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $n$: integer, $j$: integer, $z$: complex variable, $\nu$: complex parameter, $c_{2m}(q)$: coefficients and ${𝒞}_{\mu }^{(j)}$: cylinder functions

Permalink:

http://dlmf.nist.gov/28.23.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

… ►

28.23.4 $${\mathrm{me}}_{\nu }(\frac{1}{2}\pi ,h^2){\mathrm{M}}_{\nu }^{(j)}(z,h)={\mathrm{e}}^{\mathrm{i}\nu \pi /2}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_{2n}^{\nu }(h^2){𝒞}_{\nu +2n}^{(j)}(2h\sinh z),$$

ⓘ

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $n$: integer, $j$: integer, $z$: complex variable, $\nu$: complex parameter, $c_{2m}(q)$: coefficients and ${𝒞}_{\mu }^{(j)}$: cylinder functions

Permalink:

http://dlmf.nist.gov/28.23.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

►

28.23.5 $${\mathrm{me}}_{\nu }^{\prime }(\frac{1}{2}\pi ,h^2){\mathrm{M}}_{\nu }^{(j)}(z,h)=\mathrm{i}{\mathrm{e}}^{\mathrm{i}\nu \pi /2}\coth z\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}(\nu +2n)c_{2n}^{\nu }(h^2){𝒞}_{\nu +2n}^{(j)}(2h\sinh z),$$

ⓘ

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $n$: integer, $j$: integer, $z$: complex variable, $\nu$: complex parameter, $c_{2m}(q)$: coefficients and ${𝒞}_{\mu }^{(j)}$: cylinder functions

Permalink:

http://dlmf.nist.gov/28.23.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

…

9: 19.7 Connection Formulas

… ►This dichotomy of signs (missing in several references) is due to Fettis (1970). … ►The third relation (missing from the literature of Legendre’s integrals) maps each circular region onto the other and each hyperbolic region onto the other: …

10: 13.8 Asymptotic Approximations for Large Parameters

… ►For the case $b>1$ the transformation (13.2.40) can be used. … ►

13.8.13 $$𝐌(-a,b,z)\sim {\left(z/a\right)}^{(1-b)/2}\frac{{\mathrm{e}}^{z/2}\mathrm{\Gamma }\left(1+a\right)}{\mathrm{\Gamma }\left(a+b\right)}\left(J_{b-1}\left(2\sqrt{az}\right)\sum _{s=0}^{\mathrm{\infty }}\frac{p_s(z)}{{(-a)}^s}-\sqrt{z/a}{J}_b\left(2\sqrt{az}\right)\sum _{s=0}^{\mathrm{\infty }}\frac{q_s(z)}{{(-a)}^s}\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $s$: nonnegative integer and $z$: complex variable

Referenced by:

(13.8.13)

Permalink:

http://dlmf.nist.gov/13.8.E13

Encodings:

pMML, png, TeX

Addition (effective with 1.0.11):

The whole paragraph containing this equation has been added. Expansion (13.8.13) is (10.3.58) in Temme (2015). In that reference the ${(-1)}^s$ is missing.

See also:

Annotations for §13.8(iii), §13.8 and Ch.13

►

13.8.14 $$U(-a,b,z)\sim {\left(z/a\right)}^{(1-b)/2}{\mathrm{e}}^{z/2}\mathrm{\Gamma }\left(1+a\right)\left(C_{b-1}(a,2\sqrt{az})\sum _{s=0}^{\mathrm{\infty }}\frac{p_s(z)}{{(-a)}^s}-\sqrt{z/a}{C}_b(a,2\sqrt{az})\sum _{s=0}^{\mathrm{\infty }}\frac{q_s(z)}{{(-a)}^s}\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $(a,b)$: open interval, $s$: nonnegative integer and $z$: complex variable

Referenced by:

(13.8.14)

Permalink:

http://dlmf.nist.gov/13.8.E14

Encodings:

pMML, png, TeX

Addition (effective with 1.0.11):

The whole paragraph containing this equation has been added. Expansion (13.8.14) is (10.3.68) in Temme (2015). In that reference the ${(-1)}^s$ is missing.

See also:

Annotations for §13.8(iii), §13.8 and Ch.13

… ►These results follow from Temme (2022), which can also be used to obtain more terms in the expansions. …