trigonometric expansions

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1: 4.47 Approximations

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§4.47(i) Chebyshev-Series Expansions

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2: 14.13 Trigonometric Expansions

§14.13 Trigonometric Expansions

… ►For other trigonometric expansions see Erdélyi et al. (1953a, pp. 146–147).

3: 28.26 Asymptotic Approximations for Large $q$

… ►

28.26.4 $${\mathrm{Fc}}_m(z,h)\sim 1+\frac{s}{8h{\cosh }^{2}z}+\frac{1}{2^{11}{h}^2}\left(\frac{s^4+86s^2+105}{{\cosh }^{4}z}-\frac{s^4+22s^2+57}{{\cosh }^{2}z}\right)+\frac{1}{2^{14}{h}^3}\left(-\frac{s^5+14s^3+33s}{{\cosh }^{2}z}-\frac{2s^5+124s^3+1122s}{{\cosh }^{4}z}+\frac{3s^5+290s^3+1627s}{{\cosh }^{6}z}\right)+\mathrm{\cdots },$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\cosh z$: hyperbolic cosine function, $m$: integer, $h$: parameter and $z$: complex variable

A&S Ref:

20.9.9 (in slightly different notation)

Referenced by:

§28.26(i)

Permalink:

http://dlmf.nist.gov/28.26.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

►

28.26.5 $${\mathrm{Gc}}_m(z,h)\sim \frac{\sinh z}{{\cosh }^{2}z}\left(\frac{s^2+3}{2^{5}h}+\frac{1}{2^9{h}^2}\left(s^3+3s+\frac{4s^3+44s}{{\cosh }^{2}z}\right)+\frac{1}{2^{14}{h}^3}\left(5s^4+34s^2+9-\frac{s^6-47s^4+667s^2+2835}{12{\cosh }^{2}z}+\frac{s^6+505s^4+12139s^2+10395}{12{\cosh }^{4}z}\right)\right)+\mathrm{\cdots }.$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $m$: integer, $h$: parameter and $z$: complex variable

A&S Ref:

20.9.10 (in slightly different notation)

Referenced by:

§28.26(i)

Permalink:

http://dlmf.nist.gov/28.26.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

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4: 10.19 Asymptotic Expansions for Large Order

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$$J_{\nu }\left(\nu \mathrm{sech}\alpha \right)\sim \frac{{\mathrm{e}}^{\nu (\tanh \alpha -\alpha )}}{{(2\pi \nu \tanh \alpha )}^{\frac{1}{2}}}\sum _{k=0}^{\mathrm{\infty }}\frac{U_k(\coth \alpha )}{{\nu }^k},$$

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$$Y_{\nu }\left(\nu \mathrm{sech}\alpha \right)\sim -\frac{{\mathrm{e}}^{\nu (\alpha -\tanh \alpha )}}{{(\frac{1}{2}\pi \nu \tanh \alpha )}^{\frac{1}{2}}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{U_k(\coth \alpha )}{{\nu }^k},$$

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$$J_{\nu }^{\prime }\left(\nu \mathrm{sech}\alpha \right)\sim {\left(\frac{\sinh \left(2\alpha \right)}{4\pi \nu }\right)}^{\frac{1}{2}}{\mathrm{e}}^{\nu (\tanh \alpha -\alpha )}\sum _{k=0}^{\mathrm{\infty }}\frac{V_k(\coth \alpha )}{{\nu }^k},$$

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$$Y_{\nu }^{\prime }\left(\nu \mathrm{sech}\alpha \right)\sim {\left(\frac{\sinh \left(2\alpha \right)}{\pi \nu }\right)}^{\frac{1}{2}}{\mathrm{e}}^{\nu (\alpha -\tanh \alpha )}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{V_k(\coth \alpha )}{{\nu }^k}.$$

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$$J_{\nu }^{\prime }\left(\nu \sec \beta \right)\sim {\left(\frac{\sin \left(2\beta \right)}{\pi \nu }\right)}^{\frac{1}{2}}\left(-\sin \xi \sum _{k=0}^{\mathrm{\infty }}\frac{V_{2k}(\mathrm{i}\cot \beta )}{{\nu }^{2k}}-\mathrm{i}\cos \xi \sum _{k=0}^{\mathrm{\infty }}\frac{V_{2k+1}(\mathrm{i}\cot \beta )}{{\nu }^{2k+1}}\right),$$

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5: 28.8 Asymptotic Expansions for Large $q$

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28.8.11 $$P_m(x)\sim 1+\frac{s}{2^{3}h{\cos }^{2}x}+\frac{1}{h^2}\left(\frac{s^4+86s^2+105}{2^{11}{\cos }^{4}x}-\frac{s^4+22s^2+57}{2^{11}{\cos }^{2}x}\right)+\mathrm{\cdots },$$

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Defines:

$P_m(x)$ (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion, $\cos z$: cosine function, $m$: integer, $h$: parameter and $x$: real variable

A&S Ref:

20.9.14 (in different form)

Permalink:

http://dlmf.nist.gov/28.8.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(iii), §28.8 and Ch.28

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28.8.12 $$Q_m(x)\sim \frac{\sin x}{{\cos }^{2}x}\left(\frac{1}{2^{5}h}(s^2+3)+\frac{1}{2^9{h}^2}\left(s^3+3s+\frac{4s^3+44s}{{\cos }^{2}x}\right)\right)+\mathrm{\cdots }.$$

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Defines:

$Q_m(x)$ (locally)

Symbols:

$\sim$: Poincaré asymptotic expansion, $\cos z$: cosine function, $\sin z$: sine function, $m$: integer, $h$: parameter and $x$: real variable

A&S Ref:

20.9.14 (in different form)

Permalink:

http://dlmf.nist.gov/28.8.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §28.8(iii), §28.8 and Ch.28

…

6: 6.12 Asymptotic Expansions

… ►The asymptotic expansions of $\mathrm{Si}\left(z\right)$ and $\mathrm{Ci}\left(z\right)$ are given by (6.2.19), (6.2.20), together with ►

6.12.3 $$\mathrm{f}\left(z\right)\sim \frac{1}{z}\left(1-\frac{2!}{z^2}+\frac{4!}{z^4}-\frac{6!}{z^6}+\mathrm{\cdots }\right),$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

A&S Ref:

5.2.34

Referenced by:

§6.12(ii), §6.12(ii)

Permalink:

http://dlmf.nist.gov/6.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.12(ii), §6.12 and Ch.6

►

6.12.4 $$\mathrm{g}\left(z\right)\sim \frac{1}{z^2}\left(1-\frac{3!}{z^2}+\frac{5!}{z^4}-\frac{7!}{z^6}+\mathrm{\cdots }\right),$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

A&S Ref:

5.2.35

Referenced by:

§6.12(ii), §6.12(ii)

Permalink:

http://dlmf.nist.gov/6.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.12(ii), §6.12 and Ch.6

… ►When the remainders are bounded in magnitude by $\csc \left(2|\mathrm{ph}z|\right)$ times the first neglected terms. …

7: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

§22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

… ►

22.12.13 $$2K\mathrm{cs}(2Kt,k)=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\frac{\pi }{\tan \left(\pi (t-n\tau )\right)}=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\left(\underset{M\to \mathrm{\infty }}{lim}\sum _{m=-M}^M\frac{1}{t-m-n\tau }\right).$$

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Symbols:

$\mathrm{cs}(z,k)$: Jacobian elliptic 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, $\tan z$: tangent function, $k$: modulus and $\tau$: ratio

Referenced by:

§22.12

Permalink:

http://dlmf.nist.gov/22.12.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

8: 14.20 Conical (or Mehler) Functions

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§14.20(v) Trigonometric Expansion

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9: 12.10 Uniform Asymptotic Expansions for Large Parameter

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12.10.18 $$U(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})\sim \frac{2g(\mu )}{{(1-t^2)}^{\frac{1}{4}}}\left(\cos \kappa \sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\widetilde{𝒜}}_{2s}(t)}{{\mu }^{4s}}-\sin \kappa \sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\widetilde{𝒜}}_{2s+1}(t)}{{\mu }^{4s+2}}\right),$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\cos z$: cosine function, $U(a,z)$: parabolic cylinder function, $\sin z$: sine function, $s$: nonnegative integer, $\kappa$ , ${\widetilde{𝒜}}_s(t)$: coefficients and $g(\mu )$: expansion

Permalink:

http://dlmf.nist.gov/12.10.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(iv), §12.10 and Ch.12

►

12.10.19 $$U^{\prime }(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})\sim \mu \sqrt{2}g(\mu ){(1-t^2)}^{\frac{1}{4}}\left(\sin \kappa \sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\widetilde{ℬ}}_{2s}(t)}{{\mu }^{4s}}+\cos \kappa \sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\widetilde{ℬ}}_{2s+1}(t)}{{\mu }^{4s+2}}\right),$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\cos z$: cosine function, $U(a,z)$: parabolic cylinder function, $\sin z$: sine function, $s$: nonnegative integer, $\kappa$ , ${\widetilde{ℬ}}_s(t)$: coefficients and $g(\mu )$: expansion

Permalink:

http://dlmf.nist.gov/12.10.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(iv), §12.10 and Ch.12

►

12.10.20 $$V(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})\sim \frac{2g(\mu )}{\mathrm{\Gamma }\left(\frac{1}{2}+\frac{1}{2}{\mu }^2\right){(1-t^2)}^{\frac{1}{4}}}\left(\cos \chi \sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\widetilde{𝒜}}_{2s}(t)}{{\mu }^{4s}}-\sin \chi \sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\widetilde{𝒜}}_{2s+1}(t)}{{\mu }^{4s+2}}\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\cos z$: cosine function, $V(a,z)$: parabolic cylinder function, $\sin z$: sine function, $s$: nonnegative integer, $\chi$ , ${\widetilde{𝒜}}_s(t)$: coefficients and $g(\mu )$: expansion

Permalink:

http://dlmf.nist.gov/12.10.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(iv), §12.10 and Ch.12

►

12.10.21 $$V^{\prime }(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})\sim \frac{\mu \sqrt{2}g(\mu ){(1-t^2)}^{\frac{1}{4}}}{\mathrm{\Gamma }\left(\frac{1}{2}+\frac{1}{2}{\mu }^2\right)}\left(\sin \chi \sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\widetilde{ℬ}}_{2s}(t)}{{\mu }^{4s}}+\cos \chi \sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\widetilde{ℬ}}_{2s+1}(t)}{{\mu }^{4s+2}}\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\cos z$: cosine function, $V(a,z)$: parabolic cylinder function, $\sin z$: sine function, $s$: nonnegative integer, $\chi$ , ${\widetilde{ℬ}}_s(t)$: coefficients and $g(\mu )$: expansion

Permalink:

http://dlmf.nist.gov/12.10.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(iv), §12.10 and Ch.12

…

10: 12.14 The Function $W(a,x)$

… ►Other expansions, involving $\cos \left(\frac{1}{4}{x}^2\right)$ and $\sin \left(\frac{1}{4}{x}^2\right)$, can be obtained from (12.4.3) to (12.4.6) by replacing $a$ by $-\mathrm{i}a$ and $z$ by $x{\mathrm{e}}^{\pi \mathrm{i}/4}$; see Miller (1955, p. 80), and also (12.14.15) and (12.14.16). … ►

12.14.25 $$W(\frac{1}{2}{\mu }^2,\mu t\sqrt{2})\sim \frac{2^{-\frac{1}{2}}{\mathrm{e}}^{-\frac{1}{4}\pi {\mu }^2}l(\mu )}{{(t^2-1)}^{\frac{1}{4}}}\left(\cos \sigma \sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{𝒜}_{2s}(t)}{{\mu }^{4s}}-\sin \sigma \sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{𝒜}_{2s+1}(t)}{{\mu }^{4s+2}}\right),$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $W(a,x)$: parabolic cylinder function, $\sin z$: sine function, $s$: nonnegative integer, ${𝒜}_s(t)$: coefficients, $\sigma$ and $l(\mu )$

Referenced by:

§12.14(ix)

Permalink:

http://dlmf.nist.gov/12.14.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(ix), §12.14(ix), §12.14 and Ch.12

►

12.14.26 $$W(\frac{1}{2}{\mu }^2,-\mu t\sqrt{2})\sim \frac{2^{\frac{1}{2}}{\mathrm{e}}^{\frac{1}{4}\pi {\mu }^2}l(\mu )}{{(t^2-1)}^{\frac{1}{4}}}\left(\sin \sigma \sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{𝒜}_{2s}(t)}{{\mu }^{4s}}+\cos \sigma \sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{𝒜}_{2s+1}(t)}{{\mu }^{4s+2}}\right),$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $W(a,x)$: parabolic cylinder function, $\sin z$: sine function, $s$: nonnegative integer, ${𝒜}_s(t)$: coefficients, $\sigma$ and $l(\mu )$

Referenced by:

§12.14(ix)

Permalink:

http://dlmf.nist.gov/12.14.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(ix), §12.14(ix), §12.14 and Ch.12

… ►

12.14.34 $$W(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})\sim \frac{l(\mu )}{{(t^2+1)}^{\frac{1}{4}}}\left(\cos \overline{\sigma }\sum _{s=0}^{\mathrm{\infty }}\frac{{(-1)}^s{\overline{u}}_{2s}(t)}{{(t^2+1)}^{3s}{\mu }^{4s}}-\sin \overline{\sigma }\sum _{s=0}^{\mathrm{\infty }}\frac{{(-1)}^s{\overline{u}}_{2s+1}(t)}{{(t^2+1)}^{3s+\frac{3}{2}}{\mu }^{4s+2}}\right),$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\cos z$: cosine function, $W(a,x)$: parabolic cylinder function, $\sin z$: sine function, $s$: nonnegative integer, ${\overline{u}}_s(t)$: solution, $l(\mu )$ and $\overline{\sigma }$

Permalink:

http://dlmf.nist.gov/12.14.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(ix), §12.14(ix), §12.14 and Ch.12

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12.14.35 $$W^{\prime }(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})\sim -\frac{\mu }{\sqrt{2}}l(\mu ){(t^2+1)}^{\frac{1}{4}}\left(\sin \overline{\sigma }\sum _{s=0}^{\mathrm{\infty }}\frac{{(-1)}^s{\overline{v}}_{2s}(t)}{{(t^2+1)}^{3s}{\mu }^{4s}}+\cos \overline{\sigma }\sum _{s=0}^{\mathrm{\infty }}\frac{{(-1)}^s{\overline{v}}_{2s+1}(t)}{{(t^2+1)}^{3s+\frac{3}{2}}{\mu }^{4s+2}}\right),$$

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\cos z$: cosine function, $W(a,x)$: parabolic cylinder function, $\sin z$: sine function, $s$: nonnegative integer, ${\overline{v}}_s(t)$: solution, $l(\mu )$ and $\overline{\sigma }$

Permalink:

http://dlmf.nist.gov/12.14.E35

Encodings:

pMML, png, TeX

See also:

Annotations for §12.14(ix), §12.14(ix), §12.14 and Ch.12

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