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41: 5.9 Integral Representations

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5.9.10 $$\mathrm{Ln}\mathrm{\Gamma }\left(z\right)=\left(z-\frac{1}{2}\right)\ln z-z+\frac{1}{2}\ln \left(2\pi \right)+2{\int }_0^{\mathrm{\infty }}\frac{\arctan \left(t/z\right)}{{\mathrm{e}}^{2\pi t}-1}dt,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: 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, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

6.1.50

Referenced by:

(5.9.10_1), §5.9(i), §5.9(ii), Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8)

Permalink:

http://dlmf.nist.gov/5.9.E10

Encodings:

pMML, png, TeX

Addition (effective with 1.0.10):

To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\mathrm{Ln}\mathrm{\Gamma }\left(z\right)$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }\left(z\right)$ was used, where $\ln$ is the principal branch of the logarithm.

Suggested 2015-02-13 by Philippe Spindel

See also:

Annotations for §5.9(i), §5.9(i), §5.9 and Ch.5

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42: 9.1 Special Notation

… ►Other notations that have been used are as follows: $\mathrm{Ai}\left(-x\right)$ and $\mathrm{Bi}\left(-x\right)$ for $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$ (Jeffreys (1928), later changed to $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$); $U(x)=\sqrt{\pi }\mathrm{Bi}\left(x\right)$, $V(x)=\sqrt{\pi }\mathrm{Ai}\left(x\right)$ (Fock (1945)); $A(x)=3^{-1/3}\pi \mathrm{Ai}\left(-3^{-1/3}x\right)$ (Szegő (1967, §1.81)); $e_0(x)=\pi \mathrm{Hi}(-x)$, ${\tilde{e}}_0(x)=-\pi \mathrm{Gi}(-x)$ (Tumarkin (1959)).

43: 5.11 Asymptotic Expansions

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5.11.1 $$\mathrm{Ln}\mathrm{\Gamma }\left(z\right)\sim \left(z-\frac{1}{2}\right)\ln z-z+\frac{1}{2}\ln \left(2\pi \right)+\sum _{k=1}^{\mathrm{\infty }}\frac{B_{2k}}{2k(2k-1)z^{2k-1}}$$

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Symbols:

$B_n$: Bernoulli numbers, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.1.40

Referenced by:

§5.11(i), §5.11(i), §5.11(ii), §5.11(ii), Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8)

Permalink:

http://dlmf.nist.gov/5.11.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.10):

To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\mathrm{Ln}\mathrm{\Gamma }\left(z\right)$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }\left(z\right)$ was used, where $\ln$ is the principal branch of the logarithm.

Suggested 2015-02-13 by Philippe Spindel

See also:

Annotations for §5.11(i), §5.11 and Ch.5

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5.11.3 $$\mathrm{\Gamma }\left(z\right)={\mathrm{e}}^{-z}{z}^z{\left(\frac{2\pi }{z}\right)}^{1/2}{\mathrm{\Gamma }}^{\ast }\left(z\right)\sim {\mathrm{e}}^{-z}{z}^z{\left(\frac{2\pi }{z}\right)}^{1/2}\sum _{k=0}^{\mathrm{\infty }}\frac{g_k}{z^k},$$

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Defines:

${\mathrm{\Gamma }}^{\ast }\left(z\right)$: scaled gamma function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $k$: nonnegative integer, $z$: complex variable and $g_k$: coefficients

A&S Ref:

6.1.37

Referenced by:

§10.19(i), §13.8(iv), §5.11(i), §5.11(i), §5.11(i), §5.11(ii), §5.11(ii), §5.21, §5.9(i), §8.11(ii), §8.12, (8.18.13), (8.18.13), §8.18(ii), §8.18(ii), Erratum (V1.1.7) for Additions, Erratum (V1.1.8) for Rearrangement

Permalink:

http://dlmf.nist.gov/5.11.E3

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

The definition of the scaled gamma function ${\mathrm{\Gamma }}^{\ast }\left(z\right)$ was inserted after the first equals sign.

Changes (effective with 1.0.11):

An unnecessary bracket around the sum was removed.

See also:

Annotations for §5.11(i), §5.11 and Ch.5

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5.11.7 $$\mathrm{\Gamma }\left(az+b\right)\sim \sqrt{2\pi }{\mathrm{e}}^{-az}{(az)}^{az+b-(1/2)},$$

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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, $z$: complex variable, $b$: real or complex variable and $a_k$: coefficient

A&S Ref:

6.1.39

Referenced by:

§11.6(ii), §13.21(i), §2.10(iii), §2.10(iv), §26.3(v), §33.5(iv), §5.11(i)

Permalink:

http://dlmf.nist.gov/5.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §5.11(i), §5.11(i), §5.11 and Ch.5

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5.11.8 $$\mathrm{Ln}\mathrm{\Gamma }\left(z+h\right)\sim \left(z+h-\frac{1}{2}\right)\ln z-z+\frac{1}{2}\ln \left(2\pi \right)+\sum _{k=2}^{\mathrm{\infty }}\frac{{(-1)}^k{B}_k\left(h\right)}{k(k-1)z^{k-1}},$$

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $z$: complex variable

Referenced by:

§12.10(ii), (5.11.8), §5.11(i), §5.11(ii), Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8), Erratum (V1.0.11) for Equation (5.11.8)

Permalink:

http://dlmf.nist.gov/5.11.E8

Encodings:

pMML, png, TeX

Generalization (effective with 1.0.11):

Originally $h$ in (5.11.8) was unnecessarily restricted to lie in the interval $[0,1]$. In fact, $h$ may lie anywhere in the complex plane.

Suggested 2015-02-28 by Nico Temme

Addition (effective with 1.0.10):

To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\mathrm{Ln}\mathrm{\Gamma }\left(z+h\right)$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }\left(z+h\right)$ was used, where $\ln$ is the principal branch of the logarithm.

Suggested 2015-02-13 by Philippe Spindel

See also:

Annotations for §5.11(i), §5.11(i), §5.11 and Ch.5

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5.11.9 $$|\mathrm{\Gamma }\left(x+\mathrm{i}y\right)|\sim \sqrt{2\pi }{|y|}^{x-(1/2)}{\mathrm{e}}^{-\pi |y|/2},$$

ⓘ

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, $\mathrm{i}$: imaginary unit, $x$: real variable and $y$: real variable

Referenced by:

§16.5, §2.5(i), §5.11(i)

Permalink:

http://dlmf.nist.gov/5.11.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §5.11(i), §5.11(i), §5.11 and Ch.5

…

44: 8.11 Asymptotic Approximations and Expansions

… ►If $a$ is real and $z$ ($=x$) is positive, then $R_n(a,x)$ is bounded in absolute value by the first neglected term $u_n/x^n$ and has the same sign provided that $n\ge a-1$. … ►

§8.11(iii) Large $a$, Fixed $z/a$

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§8.11(iv) Large $a$, Bounded $(x-a)/{(2a)}^{\frac{1}{2}}$

►If $x=a+{(2a)}^{\frac{1}{2}}y$ and $a\to +\mathrm{\infty }$, then … ►With $x=1$, an asymptotic expansion of $e_n(nx)/{\mathrm{e}}^{nx}$ follows from (8.11.14) and (8.11.16). …

45: 11.11 Asymptotic Expansions of Anger–Weber Functions

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§11.11(iii) Large $\nu$, Fixed $z/\nu$

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$b_0(\lambda )={\displaystyle \frac{1}{{(1-{\lambda }^2)}^{1/4}}}$ ,

►

$b_1(\lambda )={\displaystyle \frac{2+3{\lambda }^2}{12{(1-{\lambda }^2)}^{7/4}}},$

►

$b_2(\lambda )={\displaystyle \frac{4+300{\lambda }^2+81{\lambda }^4}{864{(1-{\lambda }^2)}^{13/4}}}$ .

… ►(Note that Olver’s definition of ${𝐀}_{\nu }\left(z\right)$ omits the factor $1/\pi$ in (11.10.4).) …

46: 3.11 Approximation Techniques

… ►A sufficient condition for $p_n(x)$ to be the minimax polynomial is that $\left|{ϵ}_n(x)\right|$ attains its maximum at $n+2$ distinct points in $[a,b]$ and ${ϵ}_n(x)$ changes sign at these consecutive maxima. … ►where $x_{\mathrm{\ell }}=\cos \left(\pi \mathrm{\ell }/n\right)$ and the double prime means that the first and last terms are to be halved. … ►It is denoted by ${[p/q]}_f\left(z\right)$. … ►Starting with the first column ${[n/0]}_f$, $n=0,1,2,\mathrm{\dots }$, and initializing the preceding column by ${[n/-1]}_f=\mathrm{\infty }$, $n=1,2,\mathrm{\dots }$, we can compute the lower triangular part of the table via (3.11.25). Similarly, the upper triangular part follows from the first row ${[0/n]}_f$, $n=0,1,2,\mathrm{\dots }$, by initializing ${[-1/n]}_f=0$, $n=1,2,\mathrm{\dots }$. …

47: 1.11 Zeros of Polynomials

… ►The number of positive zeros of a polynomial with real coefficients cannot exceed the number of times the coefficients change sign, and the two numbers have same parity. A similar relation holds for the changes in sign of the coefficients of $f(-z)$, and hence for the number of negative zeros of $f(z)$. … ►Both polynomials have one change of sign; hence for each polynomial there is one positive zero, one negative zero, and six complex zeros. … ►The sum and product of the roots are respectively $-b/a$ and $c/a$. … ►are $1$, ${\mathrm{e}}^{2\pi \mathrm{i}/n}$, ${\mathrm{e}}^{4\pi \mathrm{i}/n},\mathrm{\dots },{\mathrm{e}}^{(2n-2)\pi \mathrm{i}/n}$, and of $z^n+1=0$ they are ${\mathrm{e}}^{\pi \mathrm{i}/n},{\mathrm{e}}^{3\pi \mathrm{i}/n},\mathrm{\dots },{\mathrm{e}}^{(2n-1)\pi \mathrm{i}/n}$. …

48: 1.5 Calculus of Two or More Variables

… ►The function $f(x,y)$ is continuously differentiable if $f$, $\partial f/\partial x$, and $\partial f/\partial y$ are continuous, and twice-continuously differentiable if also ${\partial }^{2}f/{\partial x}^2$, ${\partial }^{2}f/{\partial y}^2$, ${\partial }^{2}f/\partial x\partial y$, and ${\partial }^{2}f/\partial y\partial x$ are continuous. … ►where $f$ and its partial derivatives on the right-hand side are evaluated at $(a,b)$, and $R_n/{({\lambda }^2+{\mu }^2)}^{n/2}\to 0$ as $(\lambda ,\mu )\to (0,0)$. … ►

Change of Order of Integration

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§1.5(vi) Jacobians and Change of Variables

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Change of Variables

…

49: 3.8 Nonlinear Equations

… ►For multiple zeros the convergence is linear, but if the multiplicity $m$ is known then quadratic convergence can be restored by multiplying the ratio $f(z_n)/f^{\prime }(z_n)$ in (3.8.4) by $m$. … ►

$f(x_0)f^{\prime \prime }(x_0)>0$ and $f^{\prime }(x)$, $f^{\prime \prime }(x)$ do not change sign between $x_0$ and $\xi$ (monotonic convergence).

►

, $f^{\prime }(x)$, $f^{\prime \prime }(x)$ do not change sign in the interval $(x_0,x_1)$, and $\xi \in [x_0,x_1]$ (monotonic convergence after the first iteration).

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3.8.12 $$z_{n+1}=z_n-\frac{f(z_n)}{f^{\prime }(z_n)-(f^{\prime \prime }(z_n)f(z_n)/(2f^{\prime }(z_n)))}.$$

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Permalink:

http://dlmf.nist.gov/3.8.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §3.8(v), §3.8 and Ch.3

… ►Then the sensitivity of a simple zero $z$ to changes in $\alpha$ is given by …

50: 19.30 Lengths of Plane Curves

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19.30.2 $$s=a{\int }_0^{\varphi }\sqrt{1-k^2{\sin }^2\theta }d\theta .$$

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus and $s$: arclength

Permalink:

http://dlmf.nist.gov/19.30.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.30(i), §19.30 and Ch.19

… ►

$k^2=1-(b^2/a^2),$

… ►As $\lambda$ increases, the eccentricity $k$ decreases and the rate of change of arclength for a fixed value of $\varphi$ is given by ►

19.30.6 $$\frac{\partial s}{\partial (1/k)}=\sqrt{a^2-b^2}F(\varphi ,k)=\sqrt{a^2-b^2}{R}_F(c-1,c-k^2,c),$$ $k^2=(a^2-b^2)/(a^2+\lambda )$, $c={\csc }^2\varphi$.

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\csc z$: cosecant function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\varphi$: real or complex argument, $k$: real or complex modulus and $s$: arclength

Referenced by:

§19.30(i)

Permalink:

http://dlmf.nist.gov/19.30.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.30(i), §19.30 and Ch.19

… ►

19.30.12 $$s=aF(\varphi ,1/\sqrt{2}),$$ $\varphi =\arcsin \sqrt{2/(q+1)}=\arccos \left(\tan \theta \right)$.

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Symbols:

$F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\arccos z$: arccosine function, $\arcsin z$: arcsine function, $\tan z$: tangent function, $\varphi$: real or complex argument, $q$ and $s$: arclength

Permalink:

http://dlmf.nist.gov/19.30.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §19.30(iii), §19.30 and Ch.19

…