Watson sum

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11: 7.12 Asymptotic Expansions

… ►

$\mathrm{erfc}z\sim {\displaystyle \frac{{\mathrm{e}}^{-z^2}}{\sqrt{\pi }}}{\displaystyle \sum _{m=0}^{\mathrm{\infty }}}{(-1)}^m{\displaystyle \frac{{\left(\frac{1}{2}\right)}_m}{z^{2m+1}}},$

… ►

7.12.2 $$\mathrm{f}\left(z\right)\sim \frac{1}{\pi z}\sum _{m=0}^{\mathrm{\infty }}{(-1)}^m\frac{{\left(\frac{1}{2}\right)}_{2m}}{{(\pi z^2/2)}^{2m}},$$

ⓘ

Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter and $z$: complex variable

A&S Ref:

7.3.27

Referenced by:

§7.12(ii), §7.12(ii), Erratum (V1.0.9) for Chapters 7, 25

Permalink:

http://dlmf.nist.gov/7.12.E2

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.9):

Previously the RHS of this equation was written as $\frac{1}{\pi z}{\sum }_{m=0}^{\mathrm{\infty }}{(-1)}^m\frac{1\cdot 3\cdot 5\mathrm{\cdots }(4m-1)}{{(\pi z^2)}^{2m}}$. We have rewritten the sum more concisely using Pochhammer’s symbol.

Suggested 2014-03-13 by Giorgos Karagounis

See also:

Annotations for §7.12(ii), §7.12 and Ch.7

►

7.12.3 $$\mathrm{g}\left(z\right)\sim \frac{1}{\pi z}\sum _{m=0}^{\mathrm{\infty }}{(-1)}^m\frac{{\left(\frac{1}{2}\right)}_{2m+1}}{{(\pi z^2/2)}^{2m+1}},$$

ⓘ

Symbols:

$\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter and $z$: complex variable

A&S Ref:

7.3.28

Referenced by:

§7.12(ii), §7.12(ii), Erratum (V1.0.9) for Chapters 7, 25

Permalink:

http://dlmf.nist.gov/7.12.E3

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.9):

Previously the RHS of this equation was written as $\frac{1}{{\pi }^2{z}^3}{\sum }_{m=0}^{\mathrm{\infty }}{(-1)}^m\frac{1\cdot 3\cdot 5\mathrm{\cdots }(4m+1)}{{(\pi z^2)}^{2m}}$ We have rewritten the sum more concisely using Pochhammer’s symbol.

Suggested 2014-03-13 by Giorgos Karagounis

See also:

Annotations for §7.12(ii), §7.12 and Ch.7

… ►

7.12.4 $$\mathrm{f}\left(z\right)=\frac{1}{\pi z}\sum _{m=0}^{n-1}{(-1)}^m\frac{{\left(\frac{1}{2}\right)}_{2m}}{{(\pi z^2/2)}^{2m}}+R_n^{(\mathrm{f})}(z),$$

ⓘ

Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable, $n$: nonnegative integer and $R_n^{(\mathrm{f})}(z)$: remainder term

A&S Ref:

7.3.27 (in different form)

Referenced by:

§7.12(ii), Erratum (V1.0.9) for Chapters 7, 25

Permalink:

http://dlmf.nist.gov/7.12.E4

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.9):

Previously the RHS of this equation was written as $\frac{1}{\pi z}{\sum }_{m=0}^{n-1}{(-1)}^m\frac{1\cdot 3\mathrm{\cdots }(4m-1)}{{(\pi z^2)}^{2m}}+R_n^{(\mathrm{f})}(z)$. We have rewritten the sum more concisely using Pochhammer’s symbol.

Suggested 2014-03-13 by Giorgos Karagounis

See also:

Annotations for §7.12(ii), §7.12 and Ch.7

►

7.12.5 $$\mathrm{g}\left(z\right)=\frac{1}{\pi z}\sum _{m=0}^{n-1}{(-1)}^m\frac{{\left(\frac{1}{2}\right)}_{2m+1}}{{(\pi z^2/2)}^{2m+1}},+R_n^{(\mathrm{g})}(z),$$

ⓘ

Symbols:

$\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable, $n$: nonnegative integer and $R_n^{(\mathrm{g})}(z)$: remainder term

A&S Ref:

7.3.28 (in different form)

Referenced by:

Erratum (V1.0.9) for Chapters 7, 25

Permalink:

http://dlmf.nist.gov/7.12.E5

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.9):

Previously the RHS of this equation was written as $\frac{1}{{\pi }^2{z}^3}{\sum }_{m=0}^{n-1}{(-1)}^m\frac{1\cdot 3\mathrm{\cdots }(4m+1)}{{(\pi z^2)}^{2m}}+R_n^{(\mathrm{g})}(z)$. We have rewritten the sum more concisely using Pochhammer’s symbol.

Suggested 2014-03-13 by Giorgos Karagounis

See also:

Annotations for §7.12(ii), §7.12 and Ch.7

…

12: 10.60 Sums

§10.60 Sums

►

§10.60(i) Addition Theorems

… ►

§10.60(ii) Duplication Formulas

… ►

§10.60(iv) Compendia

… ►See also Watson (1944, Chapters 11 and 16).

13: 10.40 Asymptotic Expansions for Large Argument

… ►

10.40.3 $$I_{\nu }^{\prime }\left(z\right)\sim \frac{{\mathrm{e}}^z}{{(2\pi z)}^{\frac{1}{2}}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{b_k(\nu )}{z^k},$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$,

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant and $b_k(\nu )$: polynomial coefficient

A&S Ref:

9.7.3

Referenced by:

§10.40(i)

Permalink:

http://dlmf.nist.gov/10.40.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(i), §10.40 and Ch.10

►

10.40.4 $$K_{\nu }^{\prime }\left(z\right)\sim -{\left(\frac{\pi }{2z}\right)}^{\frac{1}{2}}{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{b_k(\nu )}{z^k},$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$.

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant and $b_k(\nu )$: polynomial coefficient

A&S Ref:

9.7.4

Referenced by:

§10.40(i), §10.40(i), §10.67(i)

Permalink:

http://dlmf.nist.gov/10.40.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(i), §10.40 and Ch.10

… ►

10.40.5 $$I_{\nu }\left(z\right)\sim \frac{{\mathrm{e}}^z}{{(2\pi z)}^{\frac{1}{2}}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{a_k(\nu )}{z^k}\pm \mathrm{i}{\mathrm{e}}^{\pm \nu \pi \mathrm{i}}\frac{{\mathrm{e}}^{-z}}{{(2\pi z)}^{\frac{1}{2}}}\sum _{k=0}^{\mathrm{\infty }}\frac{a_k(\nu )}{z^k},$$ $-\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}z\le \frac{3}{2}\pi -\delta$.

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant and $a_k(\nu )$: polynomial coefficient

Referenced by:

§10.40(i)

Permalink:

http://dlmf.nist.gov/10.40.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(i), §10.40 and Ch.10

… ►In the expansion (10.40.2) assume that $z>0$ and the sum is truncated when $k=\mathrm{\ell }-1$. … ►

10.40.10 $$K_{\nu }\left(z\right)={\left(\frac{\pi }{2z}\right)}^{\frac{1}{2}}{\mathrm{e}}^{-z}\left(\sum _{k=0}^{\mathrm{\ell }-1}\frac{a_k(\nu )}{z^k}+R_{\mathrm{\ell }}(\nu ,z)\right),$$ $\mathrm{\ell }=1,2,\mathrm{\dots }$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $a_k(\nu )$: polynomial coefficient and $R_{\mathrm{\ell }}(\nu ,z)$: remainder

Referenced by:

§10.40(i), §10.40(iii), §10.40(iv)

Permalink:

http://dlmf.nist.gov/10.40.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(iii), §10.40 and Ch.10

…

14: 2.3 Integrals of a Real Variable

… ►

§2.3(ii) Watson’s Lemma

… ►(In other words, differentiation of (2.3.8) with respect to the parameter $\lambda$ (or $\mu$) is legitimate.) … ►Watson’s lemma can be regarded as a special case of this result. ►For error bounds for Watson’s lemma and Laplace’s method see Boyd (1993) and Olver (1997b, Chapter 3). … ►The first result is the analog of Watson’s lemma (§2.3(ii)). …

15: 2.6 Distributional Methods

… ►Motivated by Watson’s lemma (§2.3(ii)), we substitute (2.6.2) in (2.6.1), and integrate term by term. … ►

2.6.41 $$f=\sum _{s=0}^{n-1}{a}_s{t}^{-s-\alpha }-\sum _{s=1}^n{c}_s{\delta }^{(s-1)}+f_n,$$

ⓘ

Symbols:

${\delta }_x$: Dirac delta distribution, $f(t)$: locally integrable function, $n$: nonnegative integer, $a_n$: coefficients, $f_n(t)$: remainder and $c_s$: coefficients

Permalink:

http://dlmf.nist.gov/2.6.E41

Encodings:

pMML, png, TeX

See also:

Annotations for §2.6(iii), §2.6 and Ch.2

… ►Since the function $t^{\mu }\left(\ln t-\gamma -\psi \left(\mu +1\right)\right)$ and all its derivatives are locally absolutely continuous in $(0,\mathrm{\infty })$, the distributional derivatives in the first sum in (2.6.44) can be replaced by the corresponding ordinary derivatives. …when $\alpha =1$, where … ►

2.6.57 $$f(t)h(xt)=\sum _{j=0}^{n-1}\sum _{k=0}^{n-1}{a}_j{b}_k{t}^{j+\alpha -1-k-\beta }{x}^{-k-\beta }+\sum _{j=0}^{n-1}{a}_j{t}^{j+\alpha -1}{h}_n(xt)+\sum _{k=0}^{n-1}{b}_k{x}^{-k-\beta }{t}^{-k-\beta }{f}_n(t)+f_n(t)h_n(xt).$$

ⓘ

Symbols:

$f(t)$: locally integrable function, $n$: positive integer, $a_n$: coefficients, $b_n$: coefficients, $h(x)$: function and $f_n(t)$: remainder

Referenced by:

§2.6(iv)

Permalink:

http://dlmf.nist.gov/2.6.E57

Encodings:

pMML, png, TeX

See also:

Annotations for §2.6(iv), §2.6 and Ch.2

…

16: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►

1.3.4 $$det[a_{jk}]=\sum _{\mathrm{\ell }=1}^n{a}_{j\mathrm{\ell }}{A}_{j\mathrm{\ell }}.$$

ⓘ

Symbols:

$det$: determinant, $j$: integer, $k$: integer, $n$: nonnegative integer and $A_{jk}$: cofactor

Permalink:

http://dlmf.nist.gov/1.3.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

… ►

1.3.9 $$det{[a_{jk}]}^2\le \left(\sum _{k=1}^n{a}_{1k}^2\right)\left(\sum _{k=1}^n{a}_{2k}^2\right)\mathrm{\dots }\left(\sum _{k=1}^n{a}_{nk}^2\right).$$

ⓘ

Symbols:

$det$: determinant, $j$: integer, $k$: integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/1.3.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §1.3(i), §1.3(i), §1.3 and Ch.1

… ►for every distinct pair of $j,k$, or when one of the factors ${\sum }_{k=1}^n{a}_{jk}^2$ vanishes. … ►

1.3.19 $$\sum _{j,k=-\mathrm{\infty }}^{\mathrm{\infty }}\left|a_{j,k}-{\delta }_{j,k}\right|$$

ⓘ

Symbols:

${\delta }_{j,k}$: Kronecker delta, $j$: integer, $k$: integer and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.3.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §1.3(iii), §1.3 and Ch.1

… ►For further information see Whittaker and Watson (1927, pp. 36–40) and Magnus and Winkler (1966, §2.3). …

17: 10.17 Asymptotic Expansions for Large Argument

… ►

10.17.3 $$J_{\nu }\left(z\right)\sim {\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}\left(\cos \omega \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{a_{2k}(\nu )}{z^{2k}}-\sin \omega \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{a_{2k+1}(\nu )}{z^{2k+1}}\right),$$ $|\mathrm{ph}z|\le \pi -\delta$,

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{ph}$: phase, $\sin z$: sine function, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant, $a_k(\nu )$: polynomial coefficient and $\omega$

A&S Ref:

9.2.5

Referenced by:

§10.17(i), §10.17(iii), §10.17(iv), §10.18(iii), §10.24, §10.49(i), §10.61(v), §10.7(ii)

Permalink:

http://dlmf.nist.gov/10.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.17(i), §10.17 and Ch.10

►

10.17.4 $$Y_{\nu }\left(z\right)\sim {\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}\left(\sin \omega \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{a_{2k}(\nu )}{z^{2k}}+\cos \omega \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{a_{2k+1}(\nu )}{z^{2k+1}}\right),$$ $|\mathrm{ph}z|\le \pi -\delta$,

ⓘ

Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{ph}$: phase, $\sin z$: sine function, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant, $a_k(\nu )$: polynomial coefficient and $\omega$

A&S Ref:

9.2.6

Referenced by:

§10.17(iii), §10.17(iv), §10.18(iii), §10.24, §10.7(ii), §11.6(i)

Permalink:

http://dlmf.nist.gov/10.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.17(i), §10.17 and Ch.10

… ►

10.17.9 $$J_{\nu }^{\prime }\left(z\right)\sim -{\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}\left(\sin \omega \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{b_{2k}(\nu )}{z^{2k}}+\cos \omega \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{b_{2k+1}(\nu )}{z^{2k+1}}\right),$$ $|\mathrm{ph}z|\le \pi -\delta$,

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{ph}$: phase, $\sin z$: sine function, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant, $\omega$ and $b_k(\nu )$: polynomial coefficient

A&S Ref:

9.2.11

Permalink:

http://dlmf.nist.gov/10.17.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.17(ii), §10.17 and Ch.10

… ►Then the remainder associated with the sum ${\sum }_{k=0}^{\mathrm{\ell }-1}{(-1)}^k{a}_{2k}(\nu )z^{-2k}$ does not exceed the first neglected term in absolute value and has the same sign provided that $\mathrm{\ell }\ge \max (\frac{1}{2}\nu -\frac{1}{4},1)$. Similarly for ${\sum }_{k=0}^{\mathrm{\ell }-1}{(-1)}^k{a}_{2k+1}(\nu )z^{-2k-1}$, provided that $\mathrm{\ell }\ge \max (\frac{1}{2}\nu -\frac{3}{4},1)$. …

18: 10.12 Generating Function and Associated Series

… ►

10.12.1 $${\mathrm{e}}^{\frac{1}{2}z(t-t^{-1})}=\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{t}^m{J}_m\left(z\right).$$

ⓘ

Symbols:

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

A&S Ref:

9.1.41

Referenced by:

§10.12, §10.23(iii), §10.35

Permalink:

http://dlmf.nist.gov/10.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.12 and Ch.10

… ►

$\cos \left(z\sin \theta \right)=J_0\left(z\right)+2{\displaystyle \sum _{k=1}^{\mathrm{\infty }}}{J}_{2k}\left(z\right)\cos \left(2k\theta \right),$

►

$\sin \left(z\sin \theta \right)=2{\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{J}_{2k+1}\left(z\right)\sin \left((2k+1)\theta \right),$

►

$\cos \left(z\cos \theta \right)=J_0\left(z\right)+2{\displaystyle \sum _{k=1}^{\mathrm{\infty }}}{(-1)}^k{J}_{2k}\left(z\right)\cos \left(2k\theta \right),$

►

$\sin \left(z\cos \theta \right)=2{\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{(-1)}^k{J}_{2k+1}\left(z\right)\cos \left((2k+1)\theta \right).$

…

19: 11.9 Lommel Functions

… ►

11.9.3 $$s_{\mu ,\nu }\left(z\right)=z^{\mu +1}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{z^{2k}}{a_{k+1}(\mu ,\nu )},$$

ⓘ

Defines:

$s_{\mu ,\nu }\left(z\right)$: Lommel function

Symbols:

$z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer and $a_k(\mu ,\nu )$: expansion function

Permalink:

http://dlmf.nist.gov/11.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §11.9(i), §11.9 and Ch.11

… ►For the foregoing results and further information see Watson (1944, §§10.7–10.73) and Babister (1967, §3.16). … ►

11.9.7 $$s_{\mu ,\nu }\left(z\right)=2^{\mu +1}\sum _{k=0}^{\mathrm{\infty }}\frac{(2k+\mu +1)\mathrm{\Gamma }\left(k+\mu +1\right)}{k!(2k+\mu -\nu +1)(2k+\mu +\nu +1)}{J}_{2k+\mu +1}\left(z\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $s_{\mu ,\nu }\left(z\right)$: Lommel function, $!$: factorial (as in $n!$), $z$: complex variable, $\nu$: real or complex order and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/11.9.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §11.9(ii), §11.9 and Ch.11

►

11.9.8 $$s_{\mu ,\nu }\left(z\right)=2^{(\mu +\nu -1)/2}\mathrm{\Gamma }\left(\frac{1}{2}\mu +\frac{1}{2}\nu +\frac{1}{2}\right)z^{(\mu +1-\nu )/2}\sum _{k=0}^{\mathrm{\infty }}\frac{{(\frac{1}{2}z)}^k}{k!(2k+\mu -\nu +1)}{J}_{k+\frac{1}{2}(\mu +\nu +1)}\left(z\right).$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $s_{\mu ,\nu }\left(z\right)$: Lommel function, $!$: factorial (as in $n!$), $z$: complex variable, $\nu$: real or complex order and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/11.9.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §11.9(ii), §11.9 and Ch.11

… ►For further information on Lommel functions see Watson (1944, §§10.7–10.75) and Babister (1967, Chapter 3). …

20: 20.2 Definitions and Periodic Properties

… ►

20.2.1 $${\theta }_1\left(z\right|\tau )={\theta }_1(z,q)=2\sum _{n=0}^{\mathrm{\infty }}{(-1)}^n{q}^{{(n+\frac{1}{2})}^2}\sin \left((2n+1)z\right),$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, ${\theta }_j\left(z\right|\tau )$: theta function, $\sin z$: sine function, $n$: integer, $z$: complex, $\tau$: lattice parameter and $q$: nome

A&S Ref:

16.27.1

Referenced by:

§20.14, §20.2(i)

Permalink:

http://dlmf.nist.gov/20.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.2(i), §20.2 and Ch.20

►

20.2.2 $${\theta }_2\left(z\right|\tau )={\theta }_2(z,q)=2\sum _{n=0}^{\mathrm{\infty }}{q}^{{(n+\frac{1}{2})}^2}\cos \left((2n+1)z\right),$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, ${\theta }_j\left(z\right|\tau )$: theta function, $\cos z$: cosine function, $n$: integer, $z$: complex, $\tau$: lattice parameter and $q$: nome

A&S Ref:

16.27.2

Permalink:

http://dlmf.nist.gov/20.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §20.2(i), §20.2 and Ch.20

►

20.2.3 $${\theta }_3\left(z\right|\tau )={\theta }_3(z,q)=1+2\sum _{n=1}^{\mathrm{\infty }}{q}^{n^2}\cos \left(2nz\right),$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, ${\theta }_j\left(z\right|\tau )$: theta function, $\cos z$: cosine function, $n$: integer, $z$: complex, $\tau$: lattice parameter and $q$: nome

A&S Ref:

16.27.3

Referenced by:

§20.10(i), §20.13, §20.14, §27.13(iv)

Permalink:

http://dlmf.nist.gov/20.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §20.2(i), §20.2 and Ch.20

►

20.2.4 $${\theta }_4\left(z\right|\tau )={\theta }_4(z,q)=1+2\sum _{n=1}^{\mathrm{\infty }}{(-1)}^n{q}^{n^2}\cos \left(2nz\right).$$

ⓘ

Symbols:

${\theta }_j(z,q)$: theta function, ${\theta }_j\left(z\right|\tau )$: theta function, $\cos z$: cosine function, $n$: integer, $z$: complex, $\tau$: lattice parameter and $q$: nome

A&S Ref:

16.27.4

Referenced by:

§20.13, §20.2(i)

Permalink:

http://dlmf.nist.gov/20.2.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §20.2(i), §20.2 and Ch.20

…