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31: 28.23 Expansions in Series of Bessel Functions

… ►

${𝒞}_{\mu }^{(4)}=H_{\mu }^{(2)};$

… ►valid for all $z$ when $j=1$, and for $\mathrm{\Re }z>0$ and $|\cosh z|>1$ when $j=2,3,4$. …valid for all $z$ when $j=1$, and for $\mathrm{\Re }z>0$ and $|\sinh z|>1$ when $j=2,3,4$. … ►

28.23.13 $${\mathrm{Ms}}_{2m+2}^{(j)}(z,h)={(-1)}^{m+1}{\left({\mathrm{se}}_{2m+2}^{\prime }(\frac{1}{2}\pi ,h^2)\right)}^{-1}\coth z\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}(2\mathrm{\ell }+2)B_{2\mathrm{\ell }+2}^{2m+2}(h^2){𝒞}_{2\mathrm{\ell }+2}^{(j)}(2h\sinh z).$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $B_m(q)$: Fourier coefficient

Referenced by:

§28.23

Permalink:

http://dlmf.nist.gov/28.23.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

… ►When $j=2,3,4$ the series in the even-numbered equations converge for $\mathrm{\Re }z>0$ and $|\cosh z|>1$, and the series in the odd-numbered equations converge for $\mathrm{\Re }z>0$ and $|\sinh z|>1$. …

32: 28.31 Equations of Whittaker–Hill and Ince

… ►It has been discussed in detail by Arscott (1967) for , and by Urwin and Arscott (1970) for $k^2>0$. … ►Formal $2\pi$-periodic solutions can be constructed as Fourier series; compare §28.4: … ►For $m_1\ne m_2$, …and also for all $p_1,p_2,m_1,m_2$, given by … ►All other periodic solutions behave as multiples of $\exp \left(\frac{1}{4}\xi \cos \left(2z\right)\right){(\cos z)}^{-p-2}$. …

33: 14.29 Generalizations

… ►

14.29.1 $$\left(1-z^2\right)\frac{d^{2}w}{{dz}^2}-2z\frac{dw}{dz}+\left(\nu (\nu +1)-\frac{{\mu }_1^2}{2(1-z)}-\frac{{\mu }_2^2}{2(1+z)}\right)w=0$$

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.29

Permalink:

http://dlmf.nist.gov/14.29.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.29 and Ch.14

… ►As in the case of (14.21.1), the solutions are hypergeometric functions, and (14.29.1) reduces to (14.21.1) when ${\mu }_1={\mu }_2=\mu$. …

34: 18.31 Bernstein–Szegő Polynomials

… ►The Bernstein–Szegő polynomials $\{p_n(x)\}$, $n=0,1,\mathrm{\dots }$, are orthogonal on $(-1,1)$ with respect to three types of weight function: ${(1-x^2)}^{-\frac{1}{2}}{(\rho (x))}^{-1}$, ${(1-x^2)}^{\frac{1}{2}}{(\rho (x))}^{-1}$, ${(1-x)}^{\frac{1}{2}}{(1+x)}^{-\frac{1}{2}}{(\rho (x))}^{-1}$. In consequence, $p_n(\cos \theta )$ can be given explicitly in terms of $\rho (\cos \theta )$ and sines and cosines, provided that in the first case, in the second case, and in the third case. …

35: 4.25 Continued Fractions

… ►

4.25.1 $$\tan z=\frac{z}{1-}\frac{z^2}{3-}\frac{z^2}{5-}\frac{z^2}{7-}\mathrm{\cdots },$$ $z\ne \pm \frac{1}{2}\pi$, $\pm \frac{3}{2}\pi$, $\mathrm{\dots }$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.94

Permalink:

http://dlmf.nist.gov/4.25.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.25 and Ch.4

►

4.25.2 $$\tan \left(az\right)=\frac{a\tan z}{1+}\frac{(1-a^2){\tan }^{2}z}{3+}\frac{(4-a^2){\tan }^{2}z}{5+}\frac{(9-a^2){\tan }^{2}z}{7+}\mathrm{\cdots },$$ , $az\ne \pm \frac{1}{2}\pi ,\pm \frac{3}{2}\pi ,\mathrm{\dots }$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Re }$: real part, $\tan z$: tangent function, $a$: real or complex constant and $z$: complex variable

A&S Ref:

4.3.95

Permalink:

http://dlmf.nist.gov/4.25.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.25 and Ch.4

►

4.25.3 $$\frac{\arcsin z}{\sqrt{1-z^2}}=\frac{z}{1-}\frac{1\cdot 2z^2}{3-}\frac{1\cdot 2z^2}{5-}\frac{3\cdot 4z^2}{7-}\frac{3\cdot 4z^2}{9-}\mathrm{\cdots },$$

ⓘ

Symbols:

$\arcsin z$: arcsine function and $z$: complex variable

A&S Ref:

4.4.44

Permalink:

http://dlmf.nist.gov/4.25.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.25 and Ch.4

… ►

4.25.4 $$\arctan z=\frac{z}{1+}\frac{z^2}{3+}\frac{4z^2}{5+}\frac{9z^2}{7+}\frac{16z^2}{9+}\mathrm{\cdots },$$

ⓘ

Symbols:

$\arctan z$: arctangent function and $z$: complex variable

A&S Ref:

4.4.43

Referenced by:

§3.10(ii)

Permalink:

http://dlmf.nist.gov/4.25.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.25 and Ch.4

… ►

4.25.5 $${\mathrm{e}}^{2a\arctan \left(1/z\right)}=1+\frac{2a}{z-a+}\frac{a^2+1}{3z+}\frac{a^2+4}{5z+}\frac{a^2+9}{7z+}\mathrm{\cdots },$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $\arctan z$: arctangent function, $a$: real or complex constant and $z$: complex variable

A&S Ref:

4.2.42

Permalink:

http://dlmf.nist.gov/4.25.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.25 and Ch.4

…

36: 12.4 Power-Series Expansions

… ►where the initial values are given by (12.2.6)–(12.2.9), and $u_1(a,z)$ and $u_2(a,z)$ are the even and odd solutions of (12.2.2) given by ►

12.4.3 $$u_1(a,z)={\mathrm{e}}^{-\frac{1}{4}{z}^2}\left(1+(a+\frac{1}{2})\frac{z^2}{2!}+(a+\frac{1}{2})(a+\frac{5}{2})\frac{z^4}{4!}+\mathrm{\cdots }\right),$$

ⓘ

Defines:

$u_1(a,z)$: solution (locally)

Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.2.1 (modification of)

Referenced by:

§12.14(v), §12.7(iv)

Permalink:

http://dlmf.nist.gov/12.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

►

12.4.4 $$u_2(a,z)={\mathrm{e}}^{-\frac{1}{4}{z}^2}\left(z+(a+\frac{3}{2})\frac{z^3}{3!}+(a+\frac{3}{2})(a+\frac{7}{2})\frac{z^5}{5!}+\mathrm{\cdots }\right).$$

ⓘ

Defines:

$u_2(a,z)$: solution (locally)

Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.2.3 (modification of)

Permalink:

http://dlmf.nist.gov/12.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

… ►

12.4.5 $$u_1(a,z)={\mathrm{e}}^{\frac{1}{4}{z}^2}\left(1+(a-\frac{1}{2})\frac{z^2}{2!}+(a-\frac{1}{2})(a-\frac{5}{2})\frac{z^4}{4!}+\mathrm{\cdots }\right),$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable, $a$: real or complex parameter and $u_1(a,z)$: solution

A&S Ref:

19.2.2 (modification of)

Permalink:

http://dlmf.nist.gov/12.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

►

12.4.6 $$u_2(a,z)={\mathrm{e}}^{\frac{1}{4}{z}^2}\left(z+(a-\frac{3}{2})\frac{z^3}{3!}+(a-\frac{3}{2})(a-\frac{7}{2})\frac{z^5}{5!}+\mathrm{\cdots }\right).$$

ⓘ

Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable, $a$: real or complex parameter and $u_2(a,z)$: solution

A&S Ref:

19.2.4 (modification of)

Referenced by:

§12.14(v), §12.7(iv)

Permalink:

http://dlmf.nist.gov/12.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

…

37: 7.9 Continued Fractions

… ►

7.9.1 $$\sqrt{\pi }{\mathrm{e}}^{z^2}\mathrm{erfc}z=\frac{z}{z^2+}\frac{\frac{1}{2}}{1+}\frac{1}{z^2+}\frac{\frac{3}{2}}{1+}\frac{2}{z^2+}\mathrm{\cdots },$$ $\mathrm{\Re }z>0$,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

7.1.14 (in different form)

Referenced by:

§7.9

Permalink:

http://dlmf.nist.gov/7.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.9 and Ch.7

►

7.9.2 $$\sqrt{\pi }{\mathrm{e}}^{z^2}\mathrm{erfc}z=\frac{2z}{2z^2+1-}\frac{1\cdot 2}{2z^2+5-}\frac{3\cdot 4}{2z^2+9-}\mathrm{\cdots },$$ $\mathrm{\Re }z>0$,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§7.9

Permalink:

http://dlmf.nist.gov/7.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §7.9 and Ch.7

►

7.9.3 $$w\left(z\right)=\frac{\mathrm{i}}{\sqrt{\pi }}\frac{1}{z-}\frac{\frac{1}{2}}{z-}\frac{1}{z-}\frac{\frac{3}{2}}{z-}\frac{2}{z-}\mathrm{\cdots },$$ $\mathrm{\Im }z>0$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

7.1.15 (in different form)

Permalink:

http://dlmf.nist.gov/7.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §7.9 and Ch.7

…

38: 17.14 Constant Term Identities

… ►

17.14.1

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $k$: nonnegative integer, $q$: complex base, $j$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§17.14

Permalink:

http://dlmf.nist.gov/17.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §17.14, §17.14 and Ch.17

… ►

17.14.2 $$\sum _{n=0}^{\mathrm{\infty }}\frac{q^{n(n+1)}}{{(q^2;q^2)}_n{(-q;q^2)}_{n+1}}=\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(z^{-1}{q}^2;q^2)}_{\mathrm{\infty }}{(-q;q^2)}_{\mathrm{\infty }}{(z^{-1}q;q^2)}_{\mathrm{\infty }}}=\frac{1}{{(-q;q^2)}_{\mathrm{\infty }}}\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(z^{-1}q;q)}_{\mathrm{\infty }}}=\frac{H(q)}{{(-q;q^2)}_{\mathrm{\infty }}},$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer, $z$: complex variable and $H(q)$: LHS of (17.2.50)

Referenced by:

§17.14

Permalink:

http://dlmf.nist.gov/17.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §17.14, §17.14 and Ch.17

►

17.14.3 $$\sum _{n=0}^{\mathrm{\infty }}\frac{q^{n(n+1)}}{{(q^2;q^2)}_n{(-q;q^2)}_{n+1}}=\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(z^{-1};q^2)}_{\mathrm{\infty }}{(-q;q^2)}_{\mathrm{\infty }}{(z^{-1}q;q^2)}_{\mathrm{\infty }}}=\frac{1}{{(-q;q^2)}_{\mathrm{\infty }}}\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(z^{-1};q)}_{\mathrm{\infty }}}=\frac{G(q)}{{(-q;q^2)}_{\mathrm{\infty }}},$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer, $z$: complex variable and $G(q)$: LHS of (17.2.49)

Permalink:

http://dlmf.nist.gov/17.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §17.14, §17.14 and Ch.17

►

17.14.4 $$\sum _{n=0}^{\mathrm{\infty }}\frac{q^{n^2}}{{(q^2;q^2)}_n{(q;q^2)}_n}=\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(-z^{-1};q^2)}_{\mathrm{\infty }}{(q;q^2)}_{\mathrm{\infty }}{(z^{-1};q^2)}_{\mathrm{\infty }}}=\frac{1}{{(q;q^2)}_{\mathrm{\infty }}}\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(z^{-2};q^4)}_{\mathrm{\infty }}}=\frac{G(q^4)}{{(q;q^2)}_{\mathrm{\infty }}},$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer, $z$: complex variable and $G(q)$: LHS of (17.2.49)

Permalink:

http://dlmf.nist.gov/17.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §17.14, §17.14 and Ch.17

►

17.14.5 $$\sum _{n=0}^{\mathrm{\infty }}\frac{q^{n^2+2n}}{{(q^2;q^2)}_n{(q;q^2)}_{n+1}}=\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(-q^2{z}^{-1};q^2)}_{\mathrm{\infty }}{(q;q^2)}_{\mathrm{\infty }}{(z^{-1}{q}^2;q^2)}_{\mathrm{\infty }}}=\frac{1}{{(q;q^2)}_{\mathrm{\infty }}}\text{ coeff. of }{z}^0\text{ in }\frac{{(-zq;q^2)}_{\mathrm{\infty }}{(-z^{-1}q;q^2)}_{\mathrm{\infty }}{(q^2;q^2)}_{\mathrm{\infty }}}{{(q^4{z}^{-2};q^4)}_{\mathrm{\infty }}}=\frac{H(q^4)}{{(q;q^2)}_{\mathrm{\infty }}}.$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: complex base, $n$: nonnegative integer, $z$: complex variable and $H(q)$: LHS of (17.2.50)

Referenced by:

§17.14

Permalink:

http://dlmf.nist.gov/17.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §17.14, §17.14 and Ch.17

…

39: 28.10 Integral Equations

… ►

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 of $x$ , $\int$: 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 of $x$ , $\cosh z$: 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 of $x$ , $\int$: 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

… ►

28.10.7 $$\frac{2}{\pi }{\int }_0^{\pi /2}\sin z\sin t\sin \left(2h\cos z\cos t\right){\mathrm{se}}_{2n+2}(t,h^2)dt=-\frac{h{B}_2^{2n+2}(h^2)}{2{\mathrm{se}}_{2n+2}^{\prime }(\frac{1}{2}\pi ,h^2)}{\mathrm{se}}_{2n+2}(z,h^2),$$

ⓘ

Symbols:

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

A&S Ref:

20.7.22 (in different form)

Permalink:

http://dlmf.nist.gov/28.10.E7

Encodings:

pMML, png, TeX

See also:

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

►

28.10.8 $$\frac{2}{\pi }{\int }_0^{\pi /2}\cos z\cos t\sinh \left(2h\sin z\sin t\right){\mathrm{se}}_{2n+2}(t,h^2)dt=\frac{h{B}_2^{2n+2}(h^2)}{2{\mathrm{se}}_{2n+2}^{\prime }(0,h^2)}{\mathrm{se}}_{2n+2}(z,h^2).$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\sinh z$: 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.E8

Encodings:

pMML, png, TeX

See also:

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

…

40: 7.8 Inequalities

… ►

7.8.2 $x\ge 0$,

ⓘ

Symbols:

$𝖬\left(x\right)$: Mills’ ratio, $\pi$: the ratio of the circumference of a circle to its diameter and $x$: real variable

A&S Ref:

7.1.13

Referenced by:

§7.8

Permalink:

http://dlmf.nist.gov/7.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §7.8 and Ch.7

… ►

7.8.4 $x>-\frac{1}{2}\sqrt{2}$,

ⓘ

Symbols:

$𝖬\left(x\right)$: Mills’ ratio and $x$: real variable

Referenced by:

§7.8

Permalink:

http://dlmf.nist.gov/7.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §7.8 and Ch.7

►

7.8.5 $x\ge 0$.

ⓘ

Symbols:

$𝖬\left(x\right)$: Mills’ ratio and $x$: real variable

Referenced by:

§7.8

Permalink:

http://dlmf.nist.gov/7.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §7.8 and Ch.7

… ►

7.8.6 $a,x>0$.

ⓘ

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $x$: real variable

Referenced by:

§7.8

Permalink:

http://dlmf.nist.gov/7.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.8 and Ch.7

… ►The function $F\left(x\right)/\sqrt{1-{\mathrm{e}}^{-2x^2}}$ is strictly decreasing for $x>0$. …