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11: 6.9 Continued Fraction

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6.9.1 $$E_1\left(z\right)=\frac{{\mathrm{e}}^{-z}}{z+}\frac{1}{1+}\frac{1}{z+}\frac{2}{1+}\frac{2}{z+}\frac{3}{1+}\frac{3}{z+}\mathrm{\cdots },$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{ph}$: phase and $z$: complex variable

A&S Ref:

5.1.22 (special case of)

Referenced by:

§6.18(i)

Permalink:

http://dlmf.nist.gov/6.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.9 and Ch.6

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12: 15.19 Methods of Computation

… ►This is because the linear transformations map the pair $\{{\mathrm{e}}^{\pi \mathrm{i}/3},{\mathrm{e}}^{-\pi \mathrm{i}/3}\}$ onto itself. …

13: 10.37 Inequalities; Monotonicity

… ►If and $|\mathrm{ph}z|\le \frac{1}{2}\pi$, then …Note that previously we did mention that (10.37.1) holds for . This is definitely not the case. …

14: 1.10 Functions of a Complex Variable

… ►If $D=ℂ\setminus (-\mathrm{\infty },0]$ and $z=r{\mathrm{e}}^{\mathrm{i}\theta }$, then one branch is $\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, the other branch is $-\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, with in both cases. Similarly if $D=ℂ\setminus [0,\mathrm{\infty })$, then one branch is $\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, the other branch is $-\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, with in both cases. …

15: 19.6 Special Cases

… ►Circular and hyperbolic cases, including Cauchy principal values, are unified by using $R_C(x,y)$. …

16: 16.5 Integral Representations and Integrals

… ►In the case $p=q$ the left-hand side of (16.5.1) is an entire function, and the right-hand side supplies an integral representation valid when . In the case $p=q+1$ the right-hand side of (16.5.1) supplies the analytic continuation of the left-hand side from the open unit disk to the sector ; compare §16.2(iii). …

17: 10.41 Asymptotic Expansions for Large Order

… ►We then extend the validity of this property from $z\to \pm \mathrm{i}\mathrm{\infty }$ to $z\to \mathrm{\infty }$ in the sector $-\pi +\delta \le \mathrm{ph}z\le 2\pi -\delta$ in the case of $H_{\nu }^{(1)}\left(\nu z\right)$, and to $z\to \mathrm{\infty }$ in the sector $-2\pi +\delta \le \mathrm{ph}z\le \pi -\delta$ in the case of $H_{\nu }^{(2)}\left(\nu z\right)$. …

18: 22.5 Special Values

… ►For values of $K,K^{\prime }$ when $k^2=\frac{1}{2}$ (lemniscatic case) see §23.5(iii), and for $k^2={\mathrm{e}}^{\mathrm{i}\pi /3}$ (equianharmonic case) see §23.5(v). …

19: 15.9 Relations to Other Functions

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15.9.17 $$𝐅(\genfrac{}{}{0pt}{}{a,a+\frac{1}{2}}{c};z)=2^{c-1}{z}^{(1-c)/2}{(1-z)}^{-a+((c-1)/2)}{P}_{2a-c}^{1-c}\left(\frac{1}{\sqrt{1-z}}\right),$$ and .

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.9.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §15.9(iv), §15.9 and Ch.15

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15.9.19 $$𝐅(\genfrac{}{}{0pt}{}{a,b}{a-b+1};z)=z^{(b-a)/2}{(1-z)}^{-b}{P}_{-b}^{b-a}\left(\frac{1+z}{1-z}\right),$$ and .

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter and $b$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.9.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §15.9(iv), §15.9 and Ch.15

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15.9.21 $$𝐅(\genfrac{}{}{0pt}{}{a,1-a}{c};z)={\left(\frac{-z}{1-z}\right)}^{(1-c)/2}{P}_{-a}^{1-c}\left(1-2z\right),$$ .

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.9(iv)

Permalink:

http://dlmf.nist.gov/15.9.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §15.9(iv), §15.9 and Ch.15

►For the case see (14.3.1). … ►

15.9.24 $$K\left(k\right)=\frac{\pi }{2}F(\genfrac{}{}{0pt}{}{\frac{1}{2},\frac{1}{2}}{1};k^2),$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind and $k$: integer

Referenced by:

§15.9(v)

Permalink:

http://dlmf.nist.gov/15.9.E24

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §15.9(v), §15.9 and Ch.15

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20: 28.29 Definitions and Basic Properties

… ►The case $c=0$ is equivalent to …The solutions of period $\pi$ or $2\pi$ are exceptional in the following sense. … ►In the symmetric case $Q(z)=Q(-z)$, $w_{\text{I}}(z,\lambda )$ is an even solution and $w_{\text{II}}(z,\lambda )$ is an odd solution; compare §28.2(ii). …The cases $\nu =0$ and $\nu =1$ split into four subcases as in (28.2.21) and (28.2.22). … ►

§28.29(iii) Discriminant and Eigenvalues in the Real Case

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