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21: 7.12 Asymptotic Expansions

… ►When $|\mathrm{ph}z|\le \frac{1}{4}\pi$ the remainder terms are bounded in magnitude by the first neglected terms, and have the same sign as these terms when $\mathrm{ph}z=0$. When the remainder terms are bounded in magnitude by $\csc \left(2|\mathrm{ph}z|\right)$ times the first neglected terms. … ►When $|\mathrm{ph}z|\le \frac{1}{8}\pi$, $R_n^{(\mathrm{f})}(z)$ and $R_n^{(\mathrm{g})}(z)$ are bounded in magnitude by the first neglected terms in (7.12.2) and (7.12.3), respectively, and have the same signs as these terms when $\mathrm{ph}z=0$. They are bounded by $|\csc \left(4\mathrm{ph}z\right)|$ times the first neglected terms when . ►For other phase ranges use (7.4.7) and (7.4.8). …

22: 13.8 Asymptotic Approximations for Large Parameters

… ►

13.8.1 $$M(a,b,z)=\sum _{s=0}^{n-1}\frac{{\left(a\right)}_s}{{\left(b\right)}_{s}s!}{z}^s+O\left({|b|}^{-n}\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $n$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable

Referenced by:

§13.9(ii)

Permalink:

http://dlmf.nist.gov/13.8.E1

Encodings:

pMML, png, TeX

See also:

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

… ►

13.8.2 $$M(a,b,z)\sim \frac{\mathrm{\Gamma }\left(b\right)}{\mathrm{\Gamma }\left(b-a\right)}\sum _{s=0}^{\mathrm{\infty }}{\left(a\right)}_s{q}_s(z,a)b^{-s-a},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $s$: nonnegative integer and $z$: complex variable

Referenced by:

§13.8(i)

Permalink:

http://dlmf.nist.gov/13.8.E2

Encodings:

pMML, png, TeX

See also:

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

►as $b\to \mathrm{\infty }$ in $|\mathrm{ph}b|\le \pi -\delta$, where $q_0(z,a)=1$ and … ►When $a\to \mathrm{\infty }$ in $\left|\mathrm{ph}a\right|\le \pi -\delta$ and $b$ and $z$ fixed, …

23: 15.12 Asymptotic Approximations

… ►

$\mathrm{\Re }z=\frac{1}{2}$ and $\left|\mathrm{ph}c\right|\le \pi -\delta$.

… ►If , then (15.12.3) applies when $|\mathrm{ph}\lambda |\le \frac{1}{2}\pi -\delta$. … ►If , then as $\lambda \to \mathrm{\infty }$ with $|\mathrm{ph}\lambda |\le \pi -\delta$, … ►If , then as $\lambda \to \mathrm{\infty }$ with $|\mathrm{ph}\lambda |\le \pi -\delta$, … ►If , then as $\lambda \to \mathrm{\infty }$ with $|\mathrm{ph}\lambda |\le \frac{1}{2}\pi -\delta$, …

24: 15.2 Definitions and Analytical Properties

… ►

15.2.1 $$F(a,b;c;z)=\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s{\left(b\right)}_s}{{\left(c\right)}_{s}s!}{z}^s=1+\frac{ab}{c}z+\frac{a(a+1)b(b+1)}{c(c+1)2!}{z}^2+\mathrm{\cdots }=\frac{\mathrm{\Gamma }\left(c\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}\sum _{s=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(a+s\right)\mathrm{\Gamma }\left(b+s\right)}{\mathrm{\Gamma }\left(c+s\right)s!}{z}^s,$$

ⓘ

Defines:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

A&S Ref:

15.1.1

Referenced by:

§15.19(i), §15.4(iii), §18.30(v), §18.35(i), §18.35(ii), §19.19

Permalink:

http://dlmf.nist.gov/15.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.2(i), §15.2 and Ch.15

… ►The branch obtained by introducing a cut from $1$ to $+\mathrm{\infty }$ on the real $z$-axis, that is, the branch in the sector $|\mathrm{ph}\left(1-z\right)|\le \pi$, is the principal branch (or principal value) of $F(a,b;c;z)$. … ►

15.2.2 $$𝐅(a,b;c;z)=\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s{\left(b\right)}_s}{\mathrm{\Gamma }\left(c+s\right)s!}{z}^s,$$ ,

ⓘ

Defines:

$𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $s$: nonnegative integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.12(i), §16.2(v)

Permalink:

http://dlmf.nist.gov/15.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.2(i), §15.2 and Ch.15

… ►

15.2.3_5 $$\underset{c\to -n}{lim}\frac{F(a,b;c;z)}{\mathrm{\Gamma }\left(c\right)}=𝐅(a,b;-n;z)=\frac{{\left(a\right)}_{n+1}{\left(b\right)}_{n+1}}{(n+1)!}{z}^{n+1}F(a+n+1,b+n+1;n+2;z),$$ $n=0,1,2,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $n$: integer, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.2(ii), Erratum (V1.0.28) for Equations (15.2.3_5), (19.11.6_5)

Permalink:

http://dlmf.nist.gov/15.2.E3_5

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.28):

This equation was given a number.

See also:

Annotations for §15.2(ii), §15.2 and Ch.15

… ►

15.2.4 $$F(-m,b;c;z)=\sum _{n=0}^m\frac{{\left(-m\right)}_n{\left(b\right)}_n}{{\left(c\right)}_{n}n!}{z}^n=\sum _{n=0}^m{(-1)}^n\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)\frac{{\left(b\right)}_n}{{\left(c\right)}_n}{z}^n.$$

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), $n$: integer, $z$: complex variable, $b$: real or complex parameter, $c$: real or complex parameter and $m$: integer

A&S Ref:

15.4.1

Permalink:

http://dlmf.nist.gov/15.2.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §15.2(ii), §15.2 and Ch.15

…

25: 13.31 Approximations

… ►Let $a,a+1-b\ne 0,-1,-2,\mathrm{\dots }$, , ►

13.31.1 $$A_n(z)=\sum _{s=0}^n\frac{{\left(-n\right)}_s{\left(n+1\right)}_s{\left(a\right)}_s{\left(b\right)}_s}{{\left(a+1\right)}_s{\left(b+1\right)}_s{(n!)}^2}{}_3{}^{}F_3^{}(\genfrac{}{}{0pt}{}{-n+s,n+1+s,1}{1+s,a+1+s,b+1+s};-z),$$

ⓘ

Defines:

$A_n(z)$: expansion (locally)

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ 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, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $n$: nonnegative integer, $s$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.31.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.31(iii), §13.31 and Ch.13

…

26: 8.11 Asymptotic Approximations and Expansions

… ►

8.11.1 $$u_k={(-1)}^k{\left(1-a\right)}_k=(a-1)(a-2)\mathrm{\cdots }(a-k),$$

ⓘ

Defines:

$u_k$ (locally)

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $a$: parameter and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/8.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(i), §8.11 and Ch.8

… ►

8.11.3 $$R_n(a,z)=O\left(z^{-n}\right),$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$,

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable, $a$: parameter, $n$: nonnegative integer, $\delta$: small positive constant and $R_n(a,z)$: remainder

Permalink:

http://dlmf.nist.gov/8.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(i), §8.11 and Ch.8

… ►This expansion is absolutely convergent for all finite $z$, and it can also be regarded as a generalized asymptotic expansion (§2.1(v)) of $\gamma (a,z)$ as $a\to \mathrm{\infty }$ in $|\mathrm{ph}a|\le \pi -\delta$. … ►

8.11.5 $$P(a,z)\sim \frac{z^a{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(1+a\right)}\sim {(2\pi a)}^{-\frac{1}{2}}{\mathrm{e}}^{a-z}{(z/a)}^a,$$ $a\to \mathrm{\infty }$, $|\mathrm{ph}a|\le \pi -\delta$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $P(a,z)$: normalized incomplete gamma function, $\mathrm{ph}$: phase, $z$: complex variable, $a$: parameter and $\delta$: small positive constant

Referenced by:

§8.11(ii)

Permalink:

http://dlmf.nist.gov/8.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(ii), §8.11 and Ch.8

… ►

8.11.12 $$\mathrm{\Gamma }(z,z)\sim z^{z-1}{\mathrm{e}}^{-z}\left(\sqrt{\frac{\pi }{2}}{z}^{\frac{1}{2}}-\frac{1}{3}+\frac{\sqrt{2\pi }}{24z^{\frac{1}{2}}}-\frac{4}{135z}+\frac{\sqrt{2\pi }}{576z^{\frac{3}{2}}}+\frac{8}{2835z^2}+\mathrm{\dots }\right),$$ $|\mathrm{ph}z|\le 2\pi -\delta$.

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\mathrm{ph}$: phase, $z$: complex variable and $\delta$: small positive constant

A&S Ref:

6.5.35 (Modified form)

Referenced by:

§8.11(v), Firefox 3 users should upgrade, Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v)

Permalink:

http://dlmf.nist.gov/8.11.E12

Encodings:

pMML, png, TeX

Addition (effective with 1.0.14):

The sector of validity for the above equation was increased from $|\mathrm{ph}z|\le \pi -\delta$ to $|\mathrm{ph}z|\le 2\pi -\delta$.

See also:

Annotations for §8.11(v), §8.11 and Ch.8

…

27: 16.11 Asymptotic Expansions

… ►As $z\to \mathrm{\infty }$ in $|\mathrm{ph}z|\le \pi$, …Here the upper or lower signs are chosen according as $z$ lies in the upper or lower half-plane; in consequence, in the fractional powers (§4.2(iv)) of $z{e}^{\mp \pi i}$ its phases are $\mathrm{ph}z\mp \pi$, respectively. (Either sign may be used when $\mathrm{ph}z=0$ since the first term on the right-hand side becomes exponentially small compared with the second term.) … ►with the same conventions on the phases of $z{e}^{\mp \pi i}$. … ►with the same conventions on the phases of $z{e}^{\mp \pi i}$. …

28: 13.2 Definitions and Basic Properties

… ►

13.2.6 $$U(a,b,z)\sim z^{-a},$$ $z\to \mathrm{\infty }$, $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$,

ⓘ

Defines:

$U(a,b,z)$: Kummer confluent hypergeometric function

Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable and $\delta$: small positive constant

Referenced by:

§13.2(i), §13.2(iv)

Permalink:

http://dlmf.nist.gov/13.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

… ►

13.2.14 $$U(-n,b,z)={(-1)}^n{\left(b\right)}_n+O\left(z\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.2.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(iii), §13.2 and Ch.13

… ►

13.2.23 $$𝐌(a,b,z)\sim {\mathrm{e}}^z{z}^{a-b}/\mathrm{\Gamma }\left(a\right),$$ $\left|\mathrm{ph}z\right|\le \frac{1}{2}\pi -\delta$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $z$: complex variable and $\delta$: small positive constant

Permalink:

http://dlmf.nist.gov/13.2.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(iv), §13.2 and Ch.13

… ►

${\mathrm{e}}^{z}U(b-a,b,{\mathrm{e}}^{-\pi \mathrm{i}}z)$ , $-\frac{1}{2}\pi \le \mathrm{ph}z\le \frac{3}{2}\pi$,

… ►

${\mathrm{e}}^{z}U(b-a,b,{\mathrm{e}}^{\pi \mathrm{i}}z)$ , $-\frac{3}{2}\pi \le \mathrm{ph}z\le \frac{1}{2}\pi$.

…

29: 25.11 Hurwitz Zeta Function

… ►

25.11.10 $$\zeta (s,a)=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(s\right)}_n}{n!}\zeta \left(n+s\right){(1-a)}^n,$$ $s\ne 1$, .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\zeta \left(s\right)$: Riemann zeta function, $!$: factorial (as in $n!$), $n$: nonnegative integer, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Keywords:

infinite series, series representation

Proof sketch:

Derivable using Taylor’s theorem (1.10.1) with $z=a$, $z_0=1$, (25.11.17), (5.2.4), (25.11.2).

Referenced by:

§25.11(iv), §25.11(iv), Erratum (V1.0.9) for Chapters 7, 25

Permalink:

http://dlmf.nist.gov/25.11.E10

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.9):

We have rewritten the original summation ${\sum }_{n=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(n+s\right)}{n!\mathrm{\Gamma }\left(s\right)}\zeta \left(n+s\right){(1-a)}^n$ more concisely as ${\sum }_{n=0}^{\mathrm{\infty }}\frac{{\left(s\right)}_n}{n!}\zeta \left(n+s\right){(1-a)}^n$ using the Pochhammer symbol.

See also:

Annotations for §25.11(iv), §25.11 and Ch.25

… ►

25.11.37 $$\sum _{k=1}^{\mathrm{\infty }}\frac{{(-1)}^k}{k}\zeta (nk,a)=-n\ln \mathrm{\Gamma }\left(a\right)+\ln \left(\prod _{j=0}^{n-1}\mathrm{\Gamma }\left(a-{\mathrm{e}}^{(2j+1)\pi \mathrm{i}/n}\right)\right),$$ $n=2,3,4,\mathrm{\dots }$, $\mathrm{\Re }a\ge 1$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer and $a$: real or complex parameter

Keywords:

infinite series, series representation

Source:

Adamchik and Srivastava (1998, (2.12), p. 136); with the choice $\mathrm{ph}\left(-1\right)=\pi$, which is reasonable since the gamma function is single valued

Permalink:

http://dlmf.nist.gov/25.11.E37

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(xi), §25.11 and Ch.25

… ►As $a\to \mathrm{\infty }$ in the sector , with $s(\ne 1)$ and $\delta$ fixed, we have the asymptotic expansion ►

25.11.43 $$\zeta (s,a)-\frac{a^{1-s}}{s-1}-\frac{1}{2}{a}^{-s}\sim \sum _{k=1}^{\mathrm{\infty }}\frac{B_{2k}}{(2k)!}{\left(s\right)}_{2k-1}{a}^{1-s-2k}.$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $k$: nonnegative integer, $a$: real or complex parameter and $s$: complex variable

Keywords:

asymptotic approximation

Source:

Paris (2005b, (1.3), (1.4), p. 298)

Referenced by:

§25.11(xii), Erratum (V1.0.9) for Chapters 7, 25

Permalink:

http://dlmf.nist.gov/25.11.E43

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.9):

We have rewritten the original summation ${\sum }_{k=1}^{\mathrm{\infty }}\frac{B_{2k}}{(2k)!}\frac{\mathrm{\Gamma }\left(s+2k-1\right)}{\mathrm{\Gamma }\left(s\right)}{a}^{1-s-2k}$ more concisely as ${\sum }_{k=1}^{\mathrm{\infty }}\frac{B_{2k}}{(2k)!}{\left(s\right)}_{2k-1}{a}^{1-s-2k}$ using the Pochhammer symbol.

See also:

Annotations for §25.11(xii), §25.11 and Ch.25

►Similarly, as $a\to \mathrm{\infty }$ in the sector , …

30: 6.7 Integral Representations

… ►

6.7.9 $$\mathrm{si}\left(z\right)=-{\int }_0^{\pi /2}{\mathrm{e}}^{-z\cos t}\cos \left(z\sin t\right)dt,$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{si}\left(z\right)$: sine integral, $\sin z$: sine function and $z$: complex variable

A&S Ref:

5.2.10

Permalink:

http://dlmf.nist.gov/6.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(ii), §6.7 and Ch.6

… ►

6.7.12 $$\mathrm{g}\left(z\right)+\mathrm{i}\mathrm{f}\left(z\right)={\mathrm{e}}^{-\mathrm{i}z}{\int }_z^{\mathrm{\infty }}\frac{{\mathrm{e}}^{\mathrm{i}t}}{t}dt,$$ $|\mathrm{ph}z|\le \pi .$

ⓘ

Symbols:

$\pi$: 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, $\mathrm{ph}$: phase, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.7(iii)

Permalink:

http://dlmf.nist.gov/6.7.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §6.7(iii), §6.7 and Ch.6

… ►The first integrals on the right-hand sides apply when ; the second ones when $\mathrm{\Re }z\ge 0$ and (in the case of (6.7.14)) $z\ne 0$. ►When …