even part

SearchAdvancedHelp

(0.002 seconds)

1—10 of 21 matching pages

1: 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

…

2: 1.12 Continued Fractions

… ►

§1.12(iv) Contraction and Extension

… ►If $C_n^{\prime }=C_{2n}$, $n=0,1,2,\mathrm{\dots }$, then $C^{\prime }$ is called the even part of $C$. The even part of $C$ exists iff $b_{2k}\ne 0$, $k=1,2,\mathrm{\dots }$, and up to equivalence is given by … ►and the even and odd parts of the continued fraction converge to finite values. …

3: 18.17 Integrals

… ►

18.17.41 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-ax}{\mathrm{𝐻𝑒}}_n\left(x\right)x^{z-1}dx=\mathrm{\Gamma }\left(z+n\right)a^{-n-2}{}_2{}^{}F_2^{}(\genfrac{}{}{0pt}{}{-\frac{1}{2}n,-\frac{1}{2}n+\frac{1}{2}}{-\frac{1}{2}z-\frac{1}{2}n,-\frac{1}{2}z-\frac{1}{2}n+\frac{1}{2}};-\frac{1}{2}{a}^2),$$ $\mathrm{\Re }a>0$. Also, $\mathrm{\Re }z>0$, $n$ even; $\mathrm{\Re }z>-1$, $n$ odd.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\mathrm{𝐻𝑒}}_n\left(x\right)$: Hermite polynomial, $dx$: differential, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Keywords:

Laplace transform, Mellin transform

Referenced by:

§18.17(vii)

Permalink:

http://dlmf.nist.gov/18.17.E41

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(vii), §18.17(vii), §18.17 and Ch.18

…

4: 28.23 Expansions in Series of Bessel Functions

… ►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 $|\sinh z|>1$. …

5: 20.1 Special Notation

… ► ►►►

$m$, $n$	integers.
…
$q$ $(\in ℂ)$	the nome, $q={\mathrm{e}}^{\mathrm{i}\pi \tau }$, . Since $\tau$ is not a single-valued function of $q$, it is assumed that $\tau$ is known, even when $q$ is specified. Most applications concern the rectangular case $\mathrm{\Re }\tau =0$, $\mathrm{\Im }\tau >0$, so that and $\tau$ and $q$ are uniquely related.
…

…

6: 19.38 Approximations

… ►They are valid over parts of the complex $k$ and $\varphi$ planes. The accuracy is controlled by the number of terms retained in the approximation; for real variables the number of significant figures appears to be roughly twice the number of terms retained, perhaps even for $\varphi$ near $\pi /2$ with the improvements made in the 1970 reference. …

7: 4.37 Inverse Hyperbolic Functions

… ►

4.37.11 $$\mathrm{arccosh}\left(-z\right)=\pm \pi \mathrm{i}+\mathrm{arccosh}z,$$ $\mathrm{\Im }z\gtrless 0$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{arccosh}z$: inverse hyperbolic cosine function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

4.6.12 (has an error even in the tenth printing.)

Referenced by:

§4.37(iii)

Permalink:

http://dlmf.nist.gov/4.37.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §4.37(iii), §4.37 and Ch.4

…

8: 14.17 Integrals

… ►

14.17.15 $${\int }_{-1}^1{𝖯}_{\nu }\left(x\right){𝖰}_{\nu }\left(x\right)dx=-\frac{\sin \left(2\nu \pi \right){\psi }^{\prime }\left(\nu +1\right)}{\pi (2\nu +1)},$$ $\mathrm{\Re }\nu >0$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential, $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, $\mathrm{\Re }$: real part, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $\sin z$: sine function, $x$: real variable and $\nu$: general degree

A&S Ref:

8.14.9

Permalink:

http://dlmf.nist.gov/14.17.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §14.17(iv), §14.17 and Ch.14

… ►(When $l+m+n$ is even the condition is not needed.) … ►

14.17.18 $${\int }_1^{\mathrm{\infty }}{P}_{\nu }\left(x\right)Q_{\lambda }\left(x\right)dx=\frac{1}{(\lambda -\nu )(\nu +\lambda +1)},$$ $\mathrm{\Re }\lambda >\mathrm{\Re }\nu >0$.

ⓘ

Symbols:

$dx$: differential, $\int$: integral, $\mathrm{\Re }$: real part, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, $x$: real variable and $\nu$: general degree

A&S Ref:

8.14.1

Permalink:

http://dlmf.nist.gov/14.17.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §14.17(iv), §14.17 and Ch.14

►

14.17.19 $${\int }_1^{\mathrm{\infty }}{Q}_{\nu }\left(x\right)Q_{\lambda }\left(x\right)dx=\frac{\psi \left(\lambda +1\right)-\psi \left(\nu +1\right)}{(\lambda -\nu )(\lambda +\nu +1)},$$ $\mathrm{\Re }\left(\lambda +\nu \right)>-1$, $\lambda \ne \nu$, $\lambda$ and $\nu \ne -1,-2,-3,\mathrm{\dots }$.

ⓘ

Symbols:

$dx$: differential, $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, $\mathrm{\Re }$: real part, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, $x$: real variable and $\nu$: general degree

A&S Ref:

8.14.2

Permalink:

http://dlmf.nist.gov/14.17.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §14.17(iv), §14.17 and Ch.14

►

14.17.20 $${\int }_1^{\mathrm{\infty }}{(Q_{\nu }\left(x\right))}^{2}dx=\frac{{\psi }^{\prime }\left(\nu +1\right)}{2\nu +1},$$ $\mathrm{\Re }\nu >-\frac{1}{2}$.

ⓘ

Symbols:

$dx$: differential, $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, $\mathrm{\Re }$: real part, $Q_{\nu }\left(z\right)=Q_{\nu }^0\left(z\right)$: Legendre function of the second kind, $x$: real variable and $\nu$: general degree

A&S Ref:

8.14.3

Permalink:

http://dlmf.nist.gov/14.17.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §14.17(iv), §14.17 and Ch.14

…

9: 20.2 Definitions and Periodic Properties

… ►For fixed $\tau$, each ${\theta }_j\left(z\right|\tau )$ is an entire function of $z$ with period $2\pi$; ${\theta }_1\left(z\right|\tau )$ is odd in $z$ and the others are even. For fixed $z$, each of ${\theta }_1\left(z\right|\tau )/\sin z$, ${\theta }_2\left(z\right|\tau )/\cos z$, ${\theta }_3\left(z\right|\tau )$, and ${\theta }_4\left(z\right|\tau )$ is an analytic function of $\tau$ for $\mathrm{\Im }\tau >0$, with a natural boundary $\mathrm{\Im }\tau =0$, and correspondingly, an analytic function of $q$ for with a natural boundary $\left|q\right|=1$. …

10: 9.13 Generalized Airy Functions

… ►

9.13.10

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $A_n\left(z\right)$: generalized Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: phase, $z$: complex variable, $\delta$: small positive constant, $n$: parameter, $p$: variable and $\zeta$: variable

Source:

Swanson and Headley (1967, (14)–(15))

Permalink:

http://dlmf.nist.gov/9.13.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(i), §9.13 and Ch.9

… ►

9.13.12

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $B_n\left(z\right)$: generalized Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: phase, $\sin z$: sine function, $z$: complex variable, $\delta$: small positive constant, $n$: parameter, $p$: variable and $\zeta$: variable

Source:

Swanson and Headley (1967, (17)–(18))

Permalink:

http://dlmf.nist.gov/9.13.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(i), §9.13 and Ch.9

…