real and imaginary parts

SearchAdvancedHelp

(0.005 seconds)

11—20 of 173 matching pages

11: 4.8 Identities

… ►where the integer $k$ is chosen so that $\mathrm{\Re }\left(-\mathrm{i}z\ln a\right)+2k\pi \in [-\pi ,\pi ]$. …

12: 25.4 Reflection Formulas

… ►

25.4.5 $${(-1)}^k{\zeta }^{(k)}\left(1-s\right)=\frac{2}{{(2\pi )}^s}\sum _{m=0}^k\sum _{r=0}^m\left(\genfrac{}{}{0.0pt}{}{k}{m}\right)\left(\genfrac{}{}{0.0pt}{}{m}{r}\right)\left(\mathrm{\Re }\left(c^{k-m}\right)\cos \left(\frac{1}{2}\pi s\right)+\mathrm{\Im }\left(c^{k-m}\right)\sin \left(\frac{1}{2}\pi s\right)\right){\mathrm{\Gamma }}^{(r)}\left(s\right){\zeta }^{(m-r)}\left(s\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{\Im }$: imaginary part, $\mathrm{\Re }$: real part, $\sin z$: sine function, $k$: nonnegative integer, $m$: nonnegative integer, $s$: complex variable and $c$

Keywords:

derivative

Source:

Apostol (1985a, (6), p. 224)

Notes:

See Apostol (1985a)

Referenced by:

§25.4

Permalink:

http://dlmf.nist.gov/25.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §25.4 and Ch.25

…

13: 10.14 Inequalities; Monotonicity

… ►

$$|J_{\nu }\left(x\right)|\le 1,$$ $\nu \ge 0,x\in ℝ$,

►

$$|J_{\nu }\left(x\right)|\le 2^{-\frac{1}{2}},$$ $\nu \ge 1,x\in ℝ$.

… ►

10.14.3 $$|J_n\left(z\right)|\le {\mathrm{e}}^{|\mathrm{\Im }z|},$$ $n\in \mathbb{Z}$.

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $\mathbb{Z}$: set of all integers, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.14 and Ch.10

►

10.14.4 $$|J_{\nu }\left(z\right)|\le \frac{{|\frac{1}{2}z|}^{\nu }{\mathrm{e}}^{|\mathrm{\Im }z|}}{\mathrm{\Gamma }\left(\nu +1\right)},$$ $\nu \ge -\frac{1}{2}$.

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.62 (corrected)

Permalink:

http://dlmf.nist.gov/10.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.14 and Ch.10

… ►

10.14.7 $$1\le \frac{J_{\nu }\left(\nu x\right)}{x^{\nu }{J}_{\nu }\left(\nu \right)}\le {\mathrm{e}}^{\nu (1-x)},$$ ;

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{e}$: base of natural logarithm, $x$: real variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.14 and Ch.10

…

14: 23.15 Definitions

… ►In §§23.15–23.19, $k$ and $k^{\prime }$ $(\in ℂ)$ denote the Jacobi modulus and complementary modulus, respectively, and $q={\mathrm{e}}^{\mathrm{i}\pi \tau }$ ($\mathrm{\Im }\tau >0$) denotes the nome; compare §§20.1 and 22.1. … ►A modular function $f(\tau )$ is a function of $\tau$ that is meromorphic in the half-plane $\mathrm{\Im }\tau >0$, and has the property that for all $𝒜\in \text{SL}(2,\mathbb{Z})$, or for all $𝒜$ belonging to a subgroup of SL$(2,\mathbb{Z})$, ►

23.15.5 $$f(𝒜\tau )=c_{𝒜}{(c\tau +d)}^{\mathrm{\ell }}f(\tau ),$$ $\mathrm{\Im }\tau >0$,

ⓘ

Symbols:

$\mathrm{\Im }$: imaginary part, $\tau$: complex variable, $𝒜$: bilinear transformation, $c$: integer, $d$: integer, $f(\tau )$: modular function, $c_{𝒜}$: constant and $\mathrm{\ell }$: level

Permalink:

http://dlmf.nist.gov/23.15.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §23.15(i), §23.15 and Ch.23

… ►

23.15.9 $$\eta \left(\tau \right)={\left(\frac{1}{2}{\theta }_1^{\prime }(0,q)\right)}^{1/3}={\mathrm{e}}^{\mathrm{i}\pi \tau /12}{\theta }_3\left(\frac{1}{2}\pi (1+\tau )\right|3\tau ).$$

ⓘ

Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), ${\theta }_j(z,q)$: theta function, ${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $q$: nome and $\tau$: complex variable

Referenced by:

§23.15(ii), §23.15(ii), §23.17(iii)

Permalink:

http://dlmf.nist.gov/23.15.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §23.15(ii), §23.15(ii), §23.15 and Ch.23

►In (23.15.9) the branch of the cube root is chosen to agree with the second equality; in particular, when $\tau$ lies on the positive imaginary axis the cube root is real and positive. …

15: 20.13 Physical Applications

… ►with $\kappa =-\mathrm{i}\pi /4$. ►For $\tau =\mathrm{i}t$, with $\alpha ,t,z$ real, (20.13.1) takes the form of a real-time $t$ diffusion equation …Let $z,\alpha ,t\in ℝ$. … ►

20.13.4 $$\sqrt{\frac{\pi }{4\alpha t}}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-{(n\pi +z)}^2/(4\alpha t)}={\theta }_3\left(z\right|\mathrm{i}4\alpha t/\pi ),$$

ⓘ

Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: integer, $t$: time, $\alpha$: real parameter and $z$: real variable

Permalink:

http://dlmf.nist.gov/20.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §20.13 and Ch.20

… ►In the singular limit $\mathrm{\Im }\tau \to 0+$, the functions ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195). …

16: 28.17 Stability as $x\to \pm \mathrm{\infty }$

… ►If all solutions of (28.2.1) are bounded when $x\to \pm \mathrm{\infty }$ along the real axis, then the corresponding pair of parameters $(a,q)$ is called stable. … ►For example, positive real values of $a$ with $q=0$ comprise stable pairs, as do values of $a$ and $q$ that correspond to real, but noninteger, values of $\nu$. ►However, if $\mathrm{\Im }\nu \ne 0$, then $(a,q)$ always comprises an unstable pair. …Also, all nontrivial solutions of (28.2.1) are unbounded on $ℝ$. ►For real $a$ and $q$ $(\ne 0)$ the stable regions are the open regions indicated in color in Figure 28.17.1. …

17: 28.9 Zeros

… ►There are no zeros within the strip other than those on the real and imaginary axes. …

18: 15.17 Mathematical Applications

… ►The quotient of two solutions of (15.10.1) maps the closed upper half-plane $\mathrm{\Im }z\ge 0$ conformally onto a curvilinear triangle. … ►First, as spherical functions on noncompact Riemannian symmetric spaces of rank one, but also as associated spherical functions, intertwining functions, matrix elements of SL$(2,ℝ)$, and spherical functions on certain nonsymmetric Gelfand pairs. …

19: 22.3 Graphics

… ►

►

►

Figure 22.3.25: $\mathrm{sn}(5,k)$ as a function of complex $k^2$, $-1\le \mathrm{\Re }\left(k^2\right)\le 3.5$, $-1\le \mathrm{\Im }\left(k^2\right)\le 1$. … Magnify 3D Help

►

►►

Figure 22.3.26: Density plot of $|\mathrm{sn}(5,k)|$ as a function of complex $k^2$, $-10\le \mathrm{\Re }\left(k^2\right)\le 20$, $-10\le \mathrm{\Im }\left(k^2\right)\le 10$. … Magnify

►

►►

Figure 22.3.27: Density plot of $|\mathrm{sn}(10,k)|$ as a function of complex $k^2$, $-10\le \mathrm{\Re }\left(k^2\right)\le 20$, $-10\le \mathrm{\Im }\left(k^2\right)\le 10$. … Magnify

►

►►

Figure 22.3.28: Density plot of $|\mathrm{sn}(20,k)|$ as a function of complex $k^2$, $-10\le \mathrm{\Re }\left(k^2\right)\le 20$, $-10\le \mathrm{\Im }\left(k^2\right)\le 10$. … Magnify

►

►►

Figure 22.3.29: Density plot of $|\mathrm{sn}(30,k)|$ as a function of complex $k^2$, $-10\le \mathrm{\Re }\left(k^2\right)\le 20$, $-10\le \mathrm{\Im }\left(k^2\right)\le 10$. … Magnify

20: 5.7 Series Expansions

… ►

5.7.8 $$\mathrm{\Im }\psi \left(1+\mathrm{i}y\right)=\sum _{k=1}^{\mathrm{\infty }}\frac{y}{k^2+y^2}.$$

ⓘ

Symbols:

$\psi \left(z\right)$: psi (or digamma) function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer and $y$: real variable

A&S Ref:

6.3.13

Permalink:

http://dlmf.nist.gov/5.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §5.7(ii), §5.7 and Ch.5