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11: 19.30 Lengths of Plane Curves

… ►

19.30.3 $$s/a=E(\varphi ,k)=R_F(c-1,c-k^2,c)-\frac{1}{3}{k}^2{R}_D(c-1,c-k^2,c),$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\varphi$: real or complex argument, $k$: real or complex modulus and $s$: arclength

Referenced by:

§19.30(i)

Permalink:

http://dlmf.nist.gov/19.30.E3

Encodings:

pMML, png, TeX

See also:

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

…

12: 4.24 Inverse Trigonometric Functions: Further Properties

… ►which requires $z$ $(=x+\mathrm{i}y)$ to lie between the two rectangular hyperbolas given by … ►

4.24.7 $$\frac{d}{dz}\arcsin z={(1-z^2)}^{-1/2},$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\arcsin z$: arcsine function and $z$: complex variable

A&S Ref:

4.4.52

Permalink:

http://dlmf.nist.gov/4.24.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(ii), §4.24 and Ch.4

►

4.24.8 $$\frac{d}{dz}\arccos z=-{(1-z^2)}^{-1/2},$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\arccos z$: arccosine function and $z$: complex variable

A&S Ref:

4.4.53

Permalink:

http://dlmf.nist.gov/4.24.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(ii), §4.24 and Ch.4

►

4.24.9 $$\frac{d}{dz}\arctan z=\frac{1}{1+z^2}.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\arctan z$: arctangent function and $z$: complex variable

A&S Ref:

4.4.54

Permalink:

http://dlmf.nist.gov/4.24.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(ii), §4.24 and Ch.4

… ►

4.24.12 $$\frac{d}{dz}\mathrm{arccot}z=-\frac{1}{1+z^2}.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{arccot}z$: arccotangent function and $z$: complex variable

A&S Ref:

4.4.55

Permalink:

http://dlmf.nist.gov/4.24.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §4.24(ii), §4.24 and Ch.4

…

13: 13.29 Methods of Computation

… ►In the following two examples Olver’s algorithm (§3.6(v)) can be used. … ►

13.29.2 $$y(n)=z^{-n-\mu -\frac{1}{2}}{M}_{\kappa ,n+\mu }\left(z\right),$$

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Defines:

$y(n)$: solution (locally)

Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.29.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.29(iv), §13.29(iv), §13.29 and Ch.13

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13.29.5 $$(n+a)w(n)-\left(2(n+a+1)+z-b\right)w(n+1)+(n+a-b+2)w(n+2)=0,$$

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Symbols:

$n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.29.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §13.29(iv), §13.29(iv), §13.29 and Ch.13

… ►

13.29.6 $$w(n)={\left(a\right)}_{n}U(n+a,b,z),$$

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Symbols:

$U(a,b,z)$: 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/13.29.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.29(iv), §13.29(iv), §13.29 and Ch.13

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13.29.7 $$z^{-a}=\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a-b+1\right)}_s}{s!}w(s),$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $s$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.29.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §13.29(iv), §13.29(iv), §13.29 and Ch.13

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14: 12.5 Integral Representations

… ►

12.5.1 $$U(a,z)=\frac{{\mathrm{e}}^{-\frac{1}{4}{z}^2}}{\mathrm{\Gamma }\left(\frac{1}{2}+a\right)}{\int }_0^{\mathrm{\infty }}{t}^{a-\frac{1}{2}}{\mathrm{e}}^{-\frac{1}{2}{t}^2-zt}dt,$$ $\mathrm{\Re }a>-\frac{1}{2}$ ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\mathrm{\Re }$: real part, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.5.3 (modification of)

Referenced by:

§12.5(i), §12.8(ii), §36.2(ii)

Permalink:

http://dlmf.nist.gov/12.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §12.5(i), §12.5 and Ch.12

►

12.5.2 $$U(a,z)=\frac{z{\mathrm{e}}^{-\frac{1}{4}{z}^2}}{\mathrm{\Gamma }\left(\frac{1}{4}+\frac{1}{2}a\right)}{\int }_0^{\mathrm{\infty }}{t}^{\frac{1}{2}a-\frac{3}{4}}{\mathrm{e}}^{-t}{\left(z^2+2t\right)}^{-\frac{1}{2}a-\frac{3}{4}}dt,$$ , $\mathrm{\Re }a>-\frac{1}{2}$ ,

ⓘ

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, $U(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.5.11 (modification of)

Permalink:

http://dlmf.nist.gov/12.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.5(i), §12.5 and Ch.12

… ►

12.5.4 $$U(a,z)=\sqrt{\frac{2}{\pi }}{\mathrm{e}}^{\frac{1}{4}{z}^2}{\int }_0^{\mathrm{\infty }}{t}^{-a-\frac{1}{2}}{\mathrm{e}}^{-\frac{1}{2}{t}^2}\cos \left(zt+\left(\frac{1}{2}a+\frac{1}{4}\right)\pi \right)dt,$$ .

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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, $U(a,z)$: parabolic cylinder function, $\mathrm{\Re }$: real part, $z$: complex variable and $a$: real or complex parameter

Referenced by:

§12.5(i)

Permalink:

http://dlmf.nist.gov/12.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.5(i), §12.5 and Ch.12

… ►Restrictions on $a$ are not needed in the following two representations: ►

12.5.6 $$U(a,z)=\frac{{\mathrm{e}}^{\frac{1}{4}{z}^2}}{\mathrm{i}\sqrt{2\pi }}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}{\mathrm{e}}^{-zt+\frac{1}{2}{t}^2}{t}^{-a-\frac{1}{2}}dt,$$ , $c>0$ ,

ⓘ

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, $U(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.5.4 (modification of)

Referenced by:

§12.8(ii)

Permalink:

http://dlmf.nist.gov/12.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §12.5(ii), §12.5 and Ch.12

…

15: 15.11 Riemann’s Differential Equation

… ►

15.11.4

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Symbols:

: Riemann’s $P$-symbol for solutions of the generalized hypergeometric differential equation, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $w$: solution

Permalink:

http://dlmf.nist.gov/15.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §15.11(i), §15.11 and Ch.15

… ►A conformal mapping of the extended complex plane onto itself has the form ►

15.11.5 $$t=(\kappa z+\lambda )/(\mu z+\nu ),$$

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Symbols:

$z$: complex variable, $t$: mapping, $\kappa$: constant, $\lambda$: constant, $\mu$: constant and $\nu$: constant

Permalink:

http://dlmf.nist.gov/15.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §15.11(ii), §15.11 and Ch.15

►where $\kappa$, $\lambda$, $\mu$, $\nu$ are real or complex constants such that $\kappa \nu -\lambda \mu =1$. These constants can be chosen to map any two sets of three distinct points $\{\alpha ,\beta ,\gamma \}$ and $\{\tilde{\alpha },\tilde{\beta },\tilde{\gamma }\}$ onto each other. …

16: 20.12 Mathematical Applications

… ►

§20.12(ii) Uniformization and Embedding of Complex Tori

►For the terminology and notation see McKean and Moll (1999, pp. 48–53). ►The space of complex tori $ℂ/(\mathbb{Z}+\tau \mathbb{Z})$ (that is, the set of complex numbers $z$ in which two of these numbers $z_1$ and $z_2$ are regarded as equivalent if there exist integers $m,n$ such that $z_1-z_2=m+\tau n$) is mapped into the projective space $P^3$ via the identification $z\to ({\theta }_1\left(2z\right|\tau ),{\theta }_2\left(2z\right|\tau ),{\theta }_3\left(2z\right|\tau ),{\theta }_4\left(2z\right|\tau ))$. Thus theta functions “uniformize” the complex torus. This ability to uniformize multiply-connected spaces (manifolds), or multi-sheeted functions of a complex variable (Riemann (1899), Rauch and Lebowitz (1973), Siegel (1988)) has led to applications in string theory (Green et al. (1988a, b), Krichever and Novikov (1989)), and also in statistical mechanics (Baxter (1982)). …

17: 4.3 Graphics

… ►

§4.3(ii) Complex Arguments: Conformal Maps

… ►Corresponding points share the same letters, with bars signifying complex conjugates. … ►

§4.3(iii) Complex Arguments: Surfaces

… ►

►

►

Figure 4.3.3: $\ln \left(x+\mathrm{i}y\right)$ (principal value). … Magnify 3D Help

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►

►

Figure 4.3.4: ${\mathrm{e}}^{x+\mathrm{i}y}$. Magnify 3D Help

18: 19.29 Reduction of General Elliptic Integrals

… ►and assume that the line segment with endpoints $a_{\alpha }+b_{\alpha }x$ and $a_{\alpha }+b_{\alpha }y$ lies in $ℂ\setminus (-\mathrm{\infty },0)$ for $1\le \alpha \le 4$. … ►The only cases of $I(𝐦)$ that are integrals of the first kind are the two ($h=3$ or 4) with $𝐦=\mathrm{𝟎}$. … ►The reduction of $I(𝐦)$ is carried out by a relation derived from partial fractions and by use of two recurrence relations. … ►It depends primarily on multivariate recurrence relations that replace one integral by two or more. … ►If both square roots in (19.29.22) are 0, then the indeterminacy in the two preceding equations can be removed by using (19.27.8) to evaluate the integral as $R_G(a_1{b}_2,a_2{b}_1,0)$ multiplied either by $-2/(b_1{b}_2)$ or by $-2/(a_1{a}_2)$ in the cases of (19.29.20) or (19.29.21), respectively. …

19: 10.25 Definitions

… ►

10.25.1 $$z^2\frac{d^{2}w}{{dz}^2}+z\frac{dw}{dz}-(z^2+{\nu }^2)w=0.$$

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Symbols:

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

A&S Ref:

9.6.1

Referenced by:

§10.25(ii), §10.25(iii), §10.27, §10.45, §10.74(ii)

Permalink:

http://dlmf.nist.gov/10.25.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.25(i), §10.25 and Ch.10

… ►In particular, the principal branch of $I_{\nu }\left(z\right)$ is defined in a similar way: it corresponds to the principal value of ${(\frac{1}{2}z)}^{\nu }$, is analytic in $ℂ\setminus (-\mathrm{\infty },0]$, and two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi$. … ►

10.25.3 $$K_{\nu }\left(z\right)\sim \sqrt{\pi /(2z)}{\mathrm{e}}^{-z},$$

ⓘ

Defines:

$K_{\nu }\left(z\right)$: modified Bessel function of the second kind

Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.25(ii), §10.25(iii), §10.30(ii)

Permalink:

http://dlmf.nist.gov/10.25.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.25(ii), §10.25 and Ch.10

… ►It has a branch point at $z=0$ for all $\nu \in ℂ$. The principal branch corresponds to the principal value of the square root in (10.25.3), is analytic in $ℂ\setminus (-\mathrm{\infty },0]$, and two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi$. …

20: 19.17 Graphics

… ►Because the $R$-function is homogeneous, there is no loss of generality in giving one variable the value $1$ or $-1$ (as in Figure 19.3.2). For $R_F$, $R_G$, and $R_J$, which are symmetric in $x,y,z$, we may further assume that $z$ is the largest of $x,y,z$ if the variables are real, then choose $z=1$, and consider only $0\le x\le 1$ and $0\le y\le 1$. … ►To view $R_F(0,y,1)$ and $2R_G(0,y,1)$ for complex $y$, put $y=1-k^2$, use (19.25.1), and see Figures 19.3.7–19.3.12. … ►To view $R_F(0,y,1)$ and $2R_G(0,y,1)$ for complex $y$, put $y=1-k^2$, use (19.25.1), and see Figures 19.3.7–19.3.12. …