complex conjugates

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21: 36.2 Catastrophes and Canonical Integrals

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36.2.23 $${\mathrm{\Psi }}_{2K+1}\left({𝐱}^{\prime }\right)=\overline{{\mathrm{\Psi }}_{2K+1}\left(𝐱\right)},$$ $x_{2m+1}^{\prime }=x_{2m+1}$, $x_{2m}^{\prime }=-x_{2m}$.

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Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\overline{z}$: complex conjugate, $n$: integer, $K$: codimension and $x_{\mathrm{i}}$: real parameter

Permalink:

http://dlmf.nist.gov/36.2.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(iii), §36.2 and Ch.36

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36.2.24 $${\mathrm{\Psi }}^{(\mathrm{U})}\left(x,y,z\right)=\overline{{\mathrm{\Psi }}^{(\mathrm{U})}\left(x,y,-z\right)},$$ $\mathrm{U}=\mathrm{E},\mathrm{H}$.

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Symbols:

$\overline{z}$: complex conjugate, ${\mathrm{\Psi }}^{(\mathrm{U})}\left(𝐱\right)$: umbilic canonical integral function, $y$: real parameter, $z$: real parameter and $x$: real parameter

Permalink:

http://dlmf.nist.gov/36.2.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §36.2(iii), §36.2 and Ch.36

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36.2.28 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,z\right)=\overline{{\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,-z\right)}=2\pi \sqrt{\frac{\pi z}{27}}\exp \left(\frac{2}{27}\mathrm{i}{z}^3\right)\left(J_{-1/6}\left(\frac{2}{27}{z}^3\right)+\mathrm{i}{J}_{1/6}\left(\frac{2}{27}{z}^3\right)\right),$$ $z\ge 0$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit and $z$: real parameter

Referenced by:

Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/36.2.E28

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This equation has been added. For the proof see Berry and Howls (2010).

See also:

Annotations for §36.2(iv), §36.2 and Ch.36

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36.2.29 $${\mathrm{\Psi }}^{(\mathrm{H})}\left(0,0,z\right)=\overline{{\mathrm{\Psi }}^{(\mathrm{H})}\left(0,0,-z\right)}=\frac{2^{1/3}}{\sqrt{3}}\exp \left(\frac{1}{27}\mathrm{i}{z}^3\right){\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,-\frac{z}{2^{2/3}}\right),$$ .

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Symbols:

$\overline{z}$: complex conjugate, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit and $z$: real parameter

Referenced by:

Erratum (V1.0.5) for Chapters 8, 20, 36

Permalink:

http://dlmf.nist.gov/36.2.E29

Encodings:

pMML, png, TeX

Addition (effective with 1.0.5):

This equation has been added. For the proof see Berry and Howls (2010).

See also:

Annotations for §36.2(iv), §36.2 and Ch.36

22: 36.8 Convergent Series Expansions

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36.8.5 $$f_n(\zeta ,\overline{\zeta })=c_n(\zeta )c_n(\overline{\zeta })\mathrm{Ai}\left(\zeta \right)\mathrm{Bi}\left(\overline{\zeta }\right)+c_n(\zeta )d_n(\overline{\zeta })\mathrm{Ai}\left(\zeta \right){\mathrm{Bi}}^{\prime }\left(\overline{\zeta }\right)+d_n(\zeta )c_n(\overline{\zeta }){\mathrm{Ai}}^{\prime }\left(\zeta \right)\mathrm{Bi}\left(\overline{\zeta }\right)+d_n(\zeta )d_n(\overline{\zeta }){\mathrm{Ai}}^{\prime }\left(\zeta \right){\mathrm{Bi}}^{\prime }\left(\overline{\zeta }\right),$$

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Defines:

$f_n$: function (locally)

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\overline{z}$: complex conjugate, $m$: integer, $c_n(t)$: coefficients and $d_n(t)$: coefficients

Permalink:

http://dlmf.nist.gov/36.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §36.8 and Ch.36

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23: 36.11 Leading-Order Asymptotics

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36.11.5 $${\mathrm{\Psi }}_3\left(0,y,0\right)=\overline{{\mathrm{\Psi }}_3(0,-y,0)}=\exp \left(\frac{1}{4}\mathrm{i}\pi \right)\sqrt{\pi /y}\left(1-(\mathrm{i}/\sqrt{3})\exp \left(\frac{3}{2}\mathrm{i}{(2y/5)}^{5/3}\right)+o\left(1\right)\right),$$ $y\to +\mathrm{\infty }$.

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Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $o\left(x\right)$: order less than and $y$: real parameter

Permalink:

http://dlmf.nist.gov/36.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11, §36.11 and Ch.36

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24: 1.8 Fourier Series

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1.8.6_1 $$\frac{1}{\pi }{\int }_{-\pi }^{\pi }f(x)\overline{g(x)}dx=\frac{1}{2}{a}_0\overline{a_0^{\prime }}+\sum _{n=1}^{\mathrm{\infty }}(a_n\overline{a_n^{\prime }}+b_n\overline{b_n^{\prime }}),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $dx$: differential of $x$ , $\int$: integral and $n$: nonnegative integer

Referenced by:

(1.8.13), (1.8.13), §1.8(i), §1.8(iv), Erratum (V1.2.0) Section 1.8

Permalink:

http://dlmf.nist.gov/1.8.E6_1

Encodings:

pMML, png, TeX

Rearrangement (effective with 1.2.0):

This equation, which was originally (1.8.13), was moved here.

See also:

Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

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1.8.6_2 $$\frac{1}{2\pi }{\int }_{-\pi }^{\pi }f(x)\overline{g(x)}dx=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_n\overline{c_n^{\prime }}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $dx$: differential of $x$ , $\int$: integral and $n$: nonnegative integer

Referenced by:

§1.8(i), Erratum (V1.2.0) Section 1.8

Permalink:

http://dlmf.nist.gov/1.8.E6_2

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

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25: 10.11 Analytic Continuation

… ►For complex $\nu$ replace $\nu$ by $\overline{\nu }$ on the right-hand sides.

26: 19.22 Quadratic Transformations

… ►If or , then $p_{+}$ and $p_{-}$ are complex conjugates. … ►If or , then $z_{+}$ and $z_{-}$ are complex conjugates. However, if $x$ and $y$ are complex conjugates and $z$ and $p$ are real, then the right-hand sides of all transformations in §§19.22(i) and 19.22(iii)—except (19.22.3) and (19.22.22)—are free of complex numbers and $p_{\pm }^2-p_{\mp }^2=\pm |p^2-x^2|\ne 0$. …

27: 3.11 Approximation Techniques

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3.11.37 $$\sum _{j=0}^{n-1}{\varphi }_k(x_j)\overline{{\varphi }_{\mathrm{\ell }}(x_j)}=n{\delta }_{k,\mathrm{\ell }},$$ $k,\mathrm{\ell }=0,1,\mathrm{\dots },n-1$,

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $\overline{z}$: complex conjugate and ${\varphi }_k(x)$: functions

Permalink:

http://dlmf.nist.gov/3.11.E37

Encodings:

pMML, png, TeX

See also:

Annotations for §3.11(v), §3.11(v), §3.11 and Ch.3

► ${\delta }_{k,\mathrm{\ell }}$ being Kronecker’s symbol and the bar denoting complex conjugate. … ►

3.11.39 $$a_k=\frac{1}{n}\sum _{j=0}^{n-1}{f}_j\overline{{\varphi }_k(x_j)},$$ $k=0,1,\mathrm{\dots },n-1$.

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Symbols:

$\overline{z}$: complex conjugate and ${\varphi }_k(x)$: functions

Permalink:

http://dlmf.nist.gov/3.11.E39

Encodings:

pMML, png, TeX

See also:

Annotations for §3.11(v), §3.11(v), §3.11 and Ch.3

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28: 28.12 Definitions and Basic Properties

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28.12.10 $$\overline{{\mathrm{me}}_{\nu }(z,q)}={\mathrm{me}}_{\overline{\nu }}(-\overline{z},\overline{q}).$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\overline{z}$: complex conjugate, $q=h^2$: parameter, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§28.12(ii)

Permalink:

http://dlmf.nist.gov/28.12.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §28.12(ii), §28.12 and Ch.28

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29: 31.15 Stieltjes Polynomials

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31.15.11 $${(f,g)}_{\rho }={\int }_{Q}f(z)\overline{g(z)}\rho (z)dz,$$

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Symbols:

$\overline{z}$: complex conjugate, $dx$: differential of $x$ , $\int$: integral, $(a,b)$: open interval, $z$: complex variable, $Q$: set and $\rho (z)$: weight function

Permalink:

http://dlmf.nist.gov/31.15.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(iii), §31.15 and Ch.31

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30: 1.17 Integral and Series Representations of the Dirac Delta

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1.17.25 $$\delta \left(\cos {\theta }_1-\cos {\theta }_2\right)\delta \left({\varphi }_1-{\varphi }_2\right)=\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}\sum _{m=-\mathrm{\ell }}^{\mathrm{\ell }}{Y}_{\mathrm{\ell },m}({\theta }_1,{\varphi }_1)\overline{Y_{\mathrm{\ell },m}({\theta }_2,{\varphi }_2)}.$$

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Symbols:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $\overline{z}$: complex conjugate, $\cos z$: cosine function, $Y_{l,m}(\theta ,\varphi )$: spherical harmonic and $m$: nonnegative integer

Referenced by:

§1.17(iii)

Permalink:

http://dlmf.nist.gov/1.17.E25

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.19):

As a notational clarification, the second spherical harmonic in the sum over $\mathrm{\ell }$ has been explicity marked up as a complexconjugate.

See also:

Annotations for §1.17(iii), §1.17(iii), §1.17 and Ch.1

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