cosine transform

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11: Guide to Searching the DLMF

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Table 1: Query Examples

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Query	Matching records contain
"Fourier transform" and series	both the phrase “Fourier transform” and the word “series”.
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Fourier (transform or series)	at least one of “Fourier transform” or “Fourier series”.
1/(2pi) and "Fourier transform"	both $1/(2\pi )$ and the phrase “Fourier transform”.
sin^2 +cos^2	the expression ${\sin }^2+{\cos }^2$.
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DeMoivre and cos (n theta)	both the word “DeMoivre” and the expression $\cos (n\theta )$.
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12: 10.32 Integral Representations

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10.32.14 $$K_{\nu }\left(z\right)=\frac{1}{2{\pi }^2\mathrm{i}}{\left(\frac{\pi }{2z}\right)}^{\frac{1}{2}}{\mathrm{e}}^{-z}\cos \left(\nu \pi \right){\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\mathrm{\Gamma }\left(t\right)\mathrm{\Gamma }\left(\frac{1}{2}-t-\nu \right)\mathrm{\Gamma }\left(\frac{1}{2}-t+\nu \right){(2z)}^{t}dt,$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\notin$: not an element of, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

Keywords:

Mellin transform

Referenced by:

§10.32(ii), §10.74(iii)

Permalink:

http://dlmf.nist.gov/10.32.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §10.32(ii), §10.32(ii), §10.32 and Ch.10

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13: 15.9 Relations to Other Functions

… ►The Jacobi transform is defined as ►

15.9.12 $$\tilde{f}(\lambda )={\int }_0^{\mathrm{\infty }}f(t){\varphi }_{\lambda }^{(\alpha ,\beta )}\left(t\right){(2\sinh t)}^{2\alpha +1}{(2\cosh t)}^{2\beta +1}dt,$$

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Symbols:

${\varphi }_{\lambda }^{(\alpha ,\beta )}\left(t\right)$: Jacobi function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral and $f(t)$: function

Permalink:

http://dlmf.nist.gov/15.9.E12

Encodings:

pMML, png, TeX

See also:

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

►with inverse … … ►Any hypergeometric function for which a quadratic transformation exists can be expressed in terms of associated Legendre functions or Ferrers functions. …

14: 24.7 Integral Representations

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24.7.7 $$B_{2n}\left(x\right)={(-1)}^{n+1}2n{\int }_0^{\mathrm{\infty }}\frac{\cos \left(2\pi x\right)-{\mathrm{e}}^{-2\pi t}}{\cosh \left(2\pi t\right)-\cos \left(2\pi x\right)}{t}^{2n-1}dt,$$ $n=1,2,\mathrm{\dots }$,

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\pi$: 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, $\cosh z$: hyperbolic cosine function, $\int$: integral, $n$: integer, $t$: real or complex and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §24.7(ii), §24.7 and Ch.24

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24.7.8 $$B_{2n+1}\left(x\right)={(-1)}^{n+1}(2n+1){\int }_0^{\mathrm{\infty }}\frac{\sin \left(2\pi x\right)}{\cosh \left(2\pi t\right)-\cos \left(2\pi x\right)}{t}^{2n}dt.$$

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\int$: integral, $\sin z$: sine function, $n$: integer, $t$: real or complex and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §24.7(ii), §24.7 and Ch.24

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24.7.9 $$E_{2n}\left(x\right)={(-1)}^{n}4{\int }_0^{\mathrm{\infty }}\frac{\sin \left(\pi x\right)\cosh \left(\pi t\right)}{\cosh \left(2\pi t\right)-\cos \left(2\pi x\right)}{t}^{2n}dt,$$

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Symbols:

$E_n\left(x\right)$: Euler polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\int$: integral, $\sin z$: sine function, $n$: integer, $t$: real or complex and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.7(ii), §24.7 and Ch.24

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24.7.10 $$E_{2n+1}\left(x\right)={(-1)}^{n+1}4{\int }_0^{\mathrm{\infty }}\frac{\cos \left(\pi x\right)\sinh \left(\pi t\right)}{\cosh \left(2\pi t\right)-\cos \left(2\pi x\right)}{t}^{2n+1}dt.$$

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Symbols:

$E_n\left(x\right)$: Euler polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $n$: integer, $t$: real or complex and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.7.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §24.7(ii), §24.7 and Ch.24

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24.7.11 $$B_n\left(x\right)=\frac{1}{2\pi \mathrm{i}}{\int }_{-c-\mathrm{i}\mathrm{\infty }}^{-c+\mathrm{i}\mathrm{\infty }}{(x+t)}^n{\left(\frac{\pi }{\sin \left(\pi t\right)}\right)}^{2}dt,$$ .

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $n$: integer, $t$: real or complex and $x$: real or complex

Keywords:

Mellin transform

Referenced by:

§24.7(ii)

Permalink:

http://dlmf.nist.gov/24.7.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §24.7(ii), §24.7(ii), §24.7 and Ch.24

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15: 7.7 Integral Representations

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7.7.15 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\cos \left(t^2\right)dt=\sqrt{\frac{\pi }{2}}\mathrm{f}\left(\frac{a}{\sqrt{2\pi }}\right),$$ $\mathrm{\Re }a>0$,

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\pi$: 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 and $\mathrm{\Re }$: real part

Keywords:

Laplace transform

A&S Ref:

7.4.22 (in different notation)

Referenced by:

§7.14(ii)

Permalink:

http://dlmf.nist.gov/7.7.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(ii), §7.7(ii), §7.7 and Ch.7

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16: 12.5 Integral Representations

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12.5.9 $$V(a,z)=\sqrt{\frac{2}{\pi }}\frac{{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}}{2\pi \mathrm{i}\mathrm{\Gamma }\left(\frac{1}{2}-a\right)}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\mathrm{\Gamma }\left(t\right)\mathrm{\Gamma }\left(\frac{1}{2}-a-2t\right)2^t{z}^{2t}\cos \left(\pi t\right)dt,$$ $a\ne \frac{1}{2},\frac{3}{2},\frac{5}{2},\mathrm{\dots }$, ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: 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, $\mathrm{i}$: imaginary unit, $\int$: integral, $V(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable and $a$: real or complex parameter

Keywords:

Mellin transform

A&S Ref:

19.5.13 (modification and correction of)

Referenced by:

§12.5(iii), §12.8(ii)

Permalink:

http://dlmf.nist.gov/12.5.E9

Encodings:

pMML, png, TeX

See also:

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

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17: 14.20 Conical (or Mehler) Functions

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14.20.3 $${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)=\frac{\pi {\mathrm{e}}^{-\tau \pi }\sin \left(\mu \pi \right)\sinh \left(\tau \pi \right)}{2({\cosh }^2\left(\tau \pi \right)-{\sin }^2\left(\mu \pi \right))}{𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)+\frac{\pi ({\mathrm{e}}^{-\tau \pi }{\cos }^2\left(\mu \pi \right)+\sinh \left(\tau \pi \right))}{2({\cosh }^2\left(\tau \pi \right)-{\sin }^2\left(\mu \pi \right))}{𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(-x\right).$$

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Symbols:

: Ferrers function of the first kind, ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$: conical function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $x$: real variable, $\tau$: real variable and $\mu$: general order

Referenced by:

§14.20(i), §14.20(iii), §14.20(ix)

Permalink:

http://dlmf.nist.gov/14.20.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.20(i), §14.20 and Ch.14

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14.20.9 $${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }\left(\cos \theta \right)=\frac{2}{\pi }{\int }_0^{\theta }\frac{\cosh \left(\tau \varphi \right)}{\sqrt{2(\cos \varphi -\cos \theta )}}d\varphi .$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\int$: integral, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $\tau$: real variable and : variable

A&S Ref:

8.12.3

Referenced by:

§14.20(iv), §14.20(v)

Permalink:

http://dlmf.nist.gov/14.20.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §14.20(iv), §14.20 and Ch.14

… ►From (14.20.9) or (14.20.10) it is evident that ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }\left(\cos \theta \right)$ is positive for real $\theta$. ►

§14.20(vi) Generalized Mehler–Fock Transformation

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14.20.13 $$P_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)=\frac{\cosh \left(\tau \pi \right)}{\pi }{\int }_1^{\mathrm{\infty }}\frac{P_{-\frac{1}{2}+\mathrm{i}\tau }\left(t\right)}{x+t}dt,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $x$: real variable and $\tau$: real variable

Permalink:

http://dlmf.nist.gov/14.20.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §14.20(vi), §14.20 and Ch.14

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18: Bibliography F

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Un théorème de Paley-Wiener pour la transformation de Fourier sur un espace riemannien symétrique de rang un. J. Funct. Anal. 49 (2), pp. 230–268.

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External Links:

ISSN 0022-1236, Document, Link, MathReview (Mogens Flensted-Jensen), ZentralBlatt

Referenced by:

§18.39(i).

Encodings:

BibTeX

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Calculation of elliptic integrals of the third kind by means of Gauss’ transformation. Math. Comp. 19 (89), pp. 97–104.

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External Links:

FullText, MathReview (J. E. L. Peck), ZentralBlatt

Referenced by:

§19.36(ii).

Encodings:

BibTeX

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Numerical calculation of singular integrals related to Hankel transform. Comput. Math. Appl. 21 (2-3), pp. 87–94.

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Notes:

Fortran, minimum precision 6D.

External Links:

ISSN 0898-1221, Document, MathReview, ZentralBlatt

Referenced by:

§10.77(ix).

Encodings:

BibTeX

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On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23 (11), pp. 2033–2042.

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External Links:

ISSN 0022-2488, Document, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.13(i), §32.7(i), §32.7(iii).

Encodings:

BibTeX

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On the asymptotic expansion of Mellin transforms. SIAM J. Math. Anal. 18 (1), pp. 273–282.

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External Links:

ISSN 0036-1410, Document, Link, MathReview (Archibald Brown), ZentralBlatt

Referenced by:

§2.3(vi).

Encodings:

BibTeX

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19: 19.8 Quadratic Transformations

§19.8 Quadratic Transformations

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§19.8(ii) Landen Transformations

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Descending Landen Transformation

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Ascending Landen Transformation

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§19.8(iii) Gauss Transformation

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20: 13.10 Integrals

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§13.10(ii) Laplace Transforms

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§13.10(iii) Mellin Transforms

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§13.10(iv) Fourier Transforms

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§13.10(v) Hankel Transforms

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