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21: 5.6 Inequalities

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5.6.6 $$|\mathrm{\Gamma }\left(x+\mathrm{i}y\right)|\le |\mathrm{\Gamma }\left(x\right)|,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{i}$: imaginary unit, $x$: real variable and $y$: real variable

A&S Ref:

6.1.26

Referenced by:

§5.6(ii)

Permalink:

http://dlmf.nist.gov/5.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §5.6(ii), §5.6 and Ch.5

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5.6.7 $$|\mathrm{\Gamma }\left(x+\mathrm{i}y\right)|\ge {(\mathrm{sech}\left(\pi y\right))}^{1/2}\mathrm{\Gamma }\left(x\right),$$ $x\ge \frac{1}{2}$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{sech}z$: hyperbolic secant function, $\mathrm{i}$: imaginary unit, $x$: real variable and $y$: real variable

Referenced by:

§5.6(ii)

Permalink:

http://dlmf.nist.gov/5.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §5.6(ii), §5.6 and Ch.5

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5.6.8 $$\left|\frac{\mathrm{\Gamma }\left(z+a\right)}{\mathrm{\Gamma }\left(z+b\right)}\right|\le \frac{1}{{|z|}^{b-a}}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $z$: complex variable, $a$: real or complex variable and $b$: real or complex variable

Referenced by:

§5.6(ii)

Permalink:

http://dlmf.nist.gov/5.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §5.6(ii), §5.6 and Ch.5

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5.6.9 $$|\mathrm{\Gamma }\left(z\right)|\le {(2\pi )}^{1/2}{|z|}^{x-(1/2)}{\mathrm{e}}^{-\pi |y|/2}\exp \left(\frac{1}{6}{|z|}^{-1}\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $x$: real variable, $y$: real variable and $z$: complex variable

Referenced by:

§5.6(ii)

Permalink:

http://dlmf.nist.gov/5.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §5.6(ii), §5.6 and Ch.5

22: 9.17 Methods of Computation

… ►Although the Maclaurin-series expansions of §§9.4 and 9.12(vi) converge for all finite values of $z$, they are cumbersome to use when $|z|$ is large owing to slowness of convergence and cancellation. For large $|z|$ the asymptotic expansions of §§9.7 and 9.12(viii) should be used instead. Since these expansions diverge, the accuracy they yield is limited by the magnitude of $|z|$. … ►In the case of $\mathrm{Ai}\left(z\right)$, for example, this means that in the sectors we may integrate along outward rays from the origin with initial values obtained from §9.2(ii). But when the integration has to be towards the origin, with starting values of $\mathrm{Ai}\left(z\right)$ and ${\mathrm{Ai}}^{\prime }\left(z\right)$ computed from their asymptotic expansions. …

23: 36.5 Stokes Sets

… ►For , there are two solutions $u$, provided that $|Y|>{(\frac{2}{5})}^{1/2}$. … ►The second sheet corresponds to $x>0$ and it intersects the bifurcation set (§36.4) smoothly along the line generated by $X=X_1=6.95643$, $\left|Y\right|=\left|Y_1\right|=6.81337$. For $\left|Y\right|>Y_1$ the second sheet is generated by a second solution of (36.5.6)–(36.5.9), and for it is generated by the roots of the polynomial equation … ►the intersection lines with the bifurcation set are generated by $|X|=X_2=0.45148$, $Y=Y_2=0.59693$. … ►Alternatively, when …

24: 10.40 Asymptotic Expansions for Large Argument

… ►as $z\to \mathrm{\infty }$ in $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$. … ►as $z\to \mathrm{\infty }$ in $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$. … ►Then the remainder term does not exceed the first neglected term in absolute value and has the same sign provided that $\mathrm{\ell }\ge \max (|\nu |-\frac{1}{2},1)$. … ►where $𝒱$ denotes the variational operator (§2.3(i)), and the paths of variation are subject to the condition that $|\mathrm{\Re }t|$ changes monotonically. … ►If $z\to \mathrm{\infty }$ with $|\mathrm{\ell }-2|z||$ bounded and $m$ $(\ge 0)$ fixed, then …

25: 6.3 Graphics

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Figure 6.3.3: $|E_1\left(x+\mathrm{i}y\right)|$, $-4\le x\le 4$, $-4\le y\le 4$. …Also, $|E_1\left(z\right)|\to \mathrm{\infty }$ logarithmically as $z\to 0$. Magnify 3D Help

26: 25.14 Lerch’s Transcendent

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25.14.1 $$\mathrm{\Phi }(z,s,a)\equiv \sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{{(a+n)}^s},$$ ; $\mathrm{\Re }s>1,|z|=1$.

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Defines:

$\mathrm{\Phi }(z,s,a)$: Lerch’s transcendent

Symbols:

$\equiv$: equals by definition, $\mathrm{\Re }$: real part, $n$: nonnegative integer, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Keywords:

definition, infinite series, series representation

Source:

Erdélyi et al. (1953a, (1.11.1), p. 27)

Referenced by:

(25.14.2), (25.14.3), §25.14(ii), §25.14, Erratum (V1.0.21) for Equation (25.14.1)

Permalink:

http://dlmf.nist.gov/25.14.E1

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.21):

The previous constraint $a\ne 0,-1,-2,\mathrm{\dots },$ was removed. A clarification regarding the correct constraints for Lerch’s transcendent $\mathrm{\Phi }(z,s,a)$ has been added in the text immediately below.

See also:

Annotations for §25.14(i), §25.14 and Ch.25

►If $s$ is not an integer then ; if $s$ is a positive integer then $a\ne 0,-1,-2,\mathrm{\dots }$; if $s$ is a non-positive integer then $a$ can be any complex number. … ►

25.14.3 $${\mathrm{Li}}_s\left(z\right)=z\mathrm{\Phi }(z,s,1),$$ $\mathrm{\Re }s>1$, $|z|\le 1$.

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Symbols:

$\mathrm{\Phi }(z,s,a)$: Lerch’s transcendent, ${\mathrm{Li}}_s\left(z\right)$: polylogarithm, $\mathrm{\Re }$: real part, $s$: complex variable and $z$: complex variable

Keywords:

specialization

Proof sketch:

Derivable from (25.12.10) and (25.14.1).

Referenced by:

(25.12.12)

Permalink:

http://dlmf.nist.gov/25.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.14(i), §25.14 and Ch.25

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25.14.6 $$\mathrm{\Phi }(z,s,a)=\frac{1}{2}{a}^{-s}+{\int }_0^{\mathrm{\infty }}\frac{z^x}{{(a+x)}^s}dx-2{\int }_0^{\mathrm{\infty }}\frac{\sin \left(x\ln z-s\arctan \left(x/a\right)\right)}{{(a^2+x^2)}^{s/2}({\mathrm{e}}^{2\pi x}-1)}dx,$$ $\mathrm{\Re }a>0$ if ; $\mathrm{\Re }s>1$, $\mathrm{\Re }a>0$ if $\left|z\right|=1$.

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Symbols:

$\mathrm{\Phi }(z,s,a)$: Lerch’s transcendent, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\arctan z$: arctangent function, $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $\sin z$: sine function, $x$: real variable, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Keywords:

improper integral, integral representation

Source:

Erdélyi et al. (1953a, (1.11.4), p. 28)

Notes:

For the case see Erdélyi et al. (1953a, (1.11.4), p. 28). In the case $\left|z\right|=1$ one checks that the first integral converges absolutely iff $\mathrm{\Re }s>1$.

Referenced by:

Erratum (V1.1.4) for Equation (25.14.6)

Permalink:

http://dlmf.nist.gov/25.14.E6

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.4):

The constraint which originally read “$\mathrm{\Re }s>0$ if ; $\mathrm{\Re }s>1$ if $|z|=1,\mathrm{\Re }a>0$” has been improved to be “$\mathrm{\Re }a>0$ if ; $\mathrm{\Re }s>1$, $\mathrm{\Re }a>0$ if $\left|z\right|=1$”.

Suggested 2021-08-23 by Gergő Nemes

See also:

Annotations for §25.14(ii), §25.14 and Ch.25

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27: 35.3 Multivariate Gamma and Beta Functions

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35.3.1 $${\mathrm{\Gamma }}_m\left(a\right)={\int }_{𝛀}\mathrm{etr}\left(-𝐗\right){\left|𝐗\right|}^{a-\frac{1}{2}(m+1)}d𝐗,$$ $\mathrm{\Re }\left(a\right)>\frac{1}{2}(m-1)$.

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Symbols:

$\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.3.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §35.3(i), §35.3 and Ch.35

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35.3.2 $${\mathrm{\Gamma }}_m\left(s_1,\mathrm{\dots },s_m\right)={\int }_{𝛀}\mathrm{etr}\left(-𝐗\right){\left|𝐗\right|}^{s_m-\frac{1}{2}(m+1)}\prod _{j=1}^{m-1}{|{(𝐗)}_j|}^{s_j-s_{j+1}}d𝐗,$$ $s_j\in ℂ$, $\mathrm{\Re }\left(s_j\right)>\frac{1}{2}(j-1)$, $j=1,\mathrm{\dots },m$.

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Symbols:

$ℂ$: complex plane, $\in$: element of, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $|{(𝐗)}_j|$: $j$th principal minor of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, $j$: nonnegative integer and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.3.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §35.3(i), §35.3 and Ch.35

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35.3.3 $\mathrm{\Re }\left(a\right),\mathrm{\Re }\left(b\right)>\frac{1}{2}(m-1)$.

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Defines:

${\mathrm{B}}_m(a,b)$: multivariate beta function

Symbols:

$\int$: integral, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $b$: complex variable, : less-than for matrices, $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Keywords:

definition, multivariate beta function

Permalink:

http://dlmf.nist.gov/35.3.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §35.3(i), §35.3 and Ch.35

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35.3.8 $${\mathrm{B}}_m(a,b)={\int }_{𝛀}{\left|𝐗\right|}^{a-\frac{1}{2}(m+1)}{\left|𝐈+𝐗\right|}^{-(a+b)}d𝐗,$$ $\mathrm{\Re }\left(a\right),\mathrm{\Re }\left(b\right)>\frac{1}{2}(m-1)$.

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Symbols:

$\int$: integral, ${\mathrm{B}}_m(a,b)$: multivariate beta function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, $b$: complex variable and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.3.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §35.3(ii), §35.3 and Ch.35

28: 33.21 Asymptotic Approximations for Large $|r|$

§33.21 Asymptotic Approximations for Large $|r|$

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§33.21(i) Limiting Forms

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§33.21(ii) Asymptotic Expansions

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29: 36.3 Visualizations of Canonical Integrals

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Figure 36.3.1: Modulus of Pearcey integral $|{\mathrm{\Psi }}_2\left(x,y\right)|$. …

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Figure 36.3.2: Modulus of swallowtail canonical integral function $|{\mathrm{\Psi }}_3\left(x,y,3\right)|$. …

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Figure 36.3.3: Modulus of swallowtail canonical integral function $|{\mathrm{\Psi }}_3\left(x,y,0\right)|$. …

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Figure 36.3.4: Modulus of swallowtail canonical integral function $|{\mathrm{\Psi }}_3\left(x,y,-3\right)|$. …

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Figure 36.3.5: Modulus of swallowtail canonical integral function $|{\mathrm{\Psi }}_3\left(x,y,-7.5\right)|$. …

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30: 18.14 Inequalities

… ►Let the maxima $x_{n,m}$, $m=0,1,\mathrm{\dots },n$, of $|P_n^{(\alpha ,\beta )}\left(x\right)|$ in $[-1,1]$ be arranged so that … ►

$|P_n^{(\alpha ,\beta )}\left(x_{n,0}\right)|>|P_n^{(\alpha ,\beta )}\left(x_{n,1}\right)|>\mathrm{\cdots }>|P_n^{(\alpha ,\beta )}\left(x_{n,m}\right)|,$

… ►except that when $\alpha =\beta =-\frac{1}{2}$ (Chebyshev case) $|P_n^{(\alpha ,\beta )}\left(x_{n,m}\right)|$ is constant. … ►Let the maxima $x_{n,m}$, $m=0,1,\mathrm{\dots },n-1$, of $|L_n^{(\alpha )}\left(x\right)|$ in $[0,\mathrm{\infty })$ be arranged so that … ►The successive maxima of $|H_n\left(x\right)|$ form a decreasing sequence for $x\le 0$, and an increasing sequence for $x\ge 0$. …