relations to trigonometric functions

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1: 4.28 Definitions and Periodicity

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Relations to Trigonometric Functions

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2: 18.11 Relations to Other Functions

… ►See §§18.5(i) and 18.5(iii) for relations to trigonometric functions, the hypergeometric function, and generalized hypergeometric functions. …

3: 18.5 Explicit Representations

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Chebyshev

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4: 11.10 Anger–Weber Functions

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11.10.18 $${𝐄}_{\nu }\left(z\right)=-\frac{1}{\pi }(1+\cos \left(\pi \nu \right))s_{0,\nu }\left(z\right)-\frac{\nu }{\pi }(1-\cos \left(\pi \nu \right))s_{-1,\nu }\left(z\right).$$

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Symbols:

$s_{\mu ,\nu }\left(z\right)$: Lommel function, ${𝐄}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $z$: complex variable and $\nu$: real or complex order

Referenced by:

§11.10(vi)

Permalink:

http://dlmf.nist.gov/11.10.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(vi), §11.10 and Ch.11

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5: 6.11 Relations to Other Functions

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6.11.3 $$\mathrm{g}\left(z\right)+\mathrm{i}\mathrm{f}\left(z\right)=U(1,1,-\mathrm{i}z).$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{i}$: imaginary unit, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.11, 5th item

Permalink:

http://dlmf.nist.gov/6.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.11, §6.11 and Ch.6

6: 19.5 Maclaurin and Related Expansions

§19.5 Maclaurin and Related Expansions

… ►where ${}_2{}^{}F_1^{}$ is the Gauss hypergeometric function (§§15.1 and 15.2(i)). … … ►Coefficients of terms up to ${\lambda }^{49}$ are given in Lee (1990), along with tables of fractional errors in $K\left(k\right)$ and $E\left(k\right)$, $0.1\le k^2\le 0.9999$, obtained by using 12 different truncations of (19.5.6) in (19.5.8) and (19.5.9). … ►An infinite series for $\ln K\left(k\right)$ is equivalent to the infinite product …

7: 7.11 Relations to Other Functions

§7.11 Relations to Other Functions

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Incomplete Gamma Functions and Generalized Exponential Integral

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Confluent Hypergeometric Functions

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7.11.6 $$C\left(z\right)+\mathrm{i}S\left(z\right)=zM(\frac{1}{2},\frac{3}{2},\frac{1}{2}\pi \mathrm{i}{z}^2)=z{\mathrm{e}}^{\pi \mathrm{i}{z}^2/2}M(1,\frac{3}{2},-\frac{1}{2}\pi \mathrm{i}{z}^2).$$

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Symbols:

$C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Permalink:

http://dlmf.nist.gov/7.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §7.11, §7.11 and Ch.7

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Generalized Hypergeometric Functions

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8: 7.5 Interrelations

§7.5 Interrelations

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7.5.3 $$C\left(z\right)=\frac{1}{2}+\mathrm{f}\left(z\right)\sin \left(\frac{1}{2}\pi z^2\right)-\mathrm{g}\left(z\right)\cos \left(\frac{1}{2}\pi z^2\right),$$

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $C\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function and $z$: complex variable

A&S Ref:

7.3.9

Referenced by:

§7.12(ii), §7.5

Permalink:

http://dlmf.nist.gov/7.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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7.5.4 $$S\left(z\right)=\frac{1}{2}-\mathrm{f}\left(z\right)\cos \left(\frac{1}{2}\pi z^2\right)-\mathrm{g}\left(z\right)\sin \left(\frac{1}{2}\pi z^2\right).$$

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function and $z$: complex variable

A&S Ref:

7.3.10

Referenced by:

§7.12(ii), §7.5

Permalink:

http://dlmf.nist.gov/7.5.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

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9: 29.2 Differential Equations

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§29.2(ii) Other Forms

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29.2.4 $$(1-k^2{\cos }^2\varphi )\frac{d^{2}w}{{d\varphi }^2}+k^2\cos \varphi \sin \varphi \frac{dw}{d\varphi }+(h-\nu (\nu +1)k^2{\cos }^2\varphi )w=0,$$

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Symbols:

$\cos z$: cosine function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\sin z$: sine function, $h$: real parameter, $k$: real parameter, $\nu$: real parameter and $\varphi$: change of variable

Permalink:

http://dlmf.nist.gov/29.2.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §29.2(ii), §29.2 and Ch.29

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29.2.5 $$\varphi =\frac{1}{2}\pi -\mathrm{am}(z,k).$$

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Defines:

$\varphi$: change of variable (locally)

Symbols:

$\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable and $k$: real parameter

Referenced by:

§29.15(i), §29.15(ii), §29.6(i)

Permalink:

http://dlmf.nist.gov/29.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §29.2(ii), §29.2 and Ch.29

… ►we have …For the Weierstrass function $\mathrm{\wp }$ see §23.2(ii). …

10: 10.39 Relations to Other Functions

§10.39 Relations to Other Functions

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Elementary Functions

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Parabolic Cylinder Functions

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Confluent Hypergeometric Functions

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Generalized Hypergeometric Functions and Hypergeometric Function

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