OpenMath Content Dictionary: errorFresnelInts

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http://www.openmath.org/CDs/errorFresnelInts.ocd
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errorFresnelInts.ocd
CD as XML Encoded OpenMath:
errorFresnelInts.omcd
Defines:
AssociatedLegendreP, AssociatedLegendreQ, erf, erfc, FresnelC, FresnelS, LegendreP, LegendreQ
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1/1/5000

This content dictionary contains symbols which describe the error function and Fresnel integrals


erf

The symbol erf represents the Error function, which is denoted $erf(z)$. It is defined as: $erf(z)=\frac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^2}dt$.

The definition is taken from M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [7.1.1]

Commented Mathematical property (CMP):
erf : real $\rightarrow$ real
Commented Mathematical property (CMP):
erf : complex $\rightarrow$ complex
Commented Mathematical property (CMP):
erf : symbolic $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$erf(-z)=-erf(z)$, symmetry
Commented Mathematical property (CMP):
$erf \bar{z}=\overline {erf(z)}$, symmetry
Signatures:
sts


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erfc

The symbol erfc represents the Complementary Error function, which is denoted $erfc(z)$. It is defined as: $erfc(z)=1-erf(z)$. The definition is taken from M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [7.1.2]

Commented Mathematical property (CMP):
erfc : real $\rightarrow$ real
Commented Mathematical property (CMP):
erfc : complex $\rightarrow$ complex
Commented Mathematical property (CMP):
erfc : symbolic $\rightarrow$ symbolic
Commented Mathematical property (CMP):
i^{-1}erfc(z)=\frac{2}{\sqrt{\pi}}e^{-z^2}
Signatures:
sts


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FresnelC

The symbol FresnelC represents the Fresnel integral Cosine, which is denoted by $C(z)$. This is defined by $C(z)=\int_{0}^{z} cos(\frac{\pi}{2} t^2) dt$. The definition is taken from M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [7.3.1]

Commented Mathematical property (CMP):
C : real $\rightarrow$ real
Commented Mathematical property (CMP):
C : complex $\rightarrow$ complex
Commented Mathematical property (CMP):
C : symbolic $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$C(z)=\sum_{n=0}^{\infty} \frac{(-1)^n(\pi /2)^{2n}} {(2n)!(4n+1)}z^{4n+1}$, series expansion
Commented Mathematical property (CMP):
$C(-z)=-C(z)$, symmetry
Commented Mathematical property (CMP):
$C(iz)=iC(z)$, symmetry
Commented Mathematical property (CMP):
$C(\bar{z})=\overline{C(z)}$, symmetry
Signatures:
sts


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FresnelS

The symbol FresnelS represents the Fresnel integral Sine, which is denoted by $S(z)$. This is defined by $S(z)=\int_{0}^{z} sin(\frac{\pi}{2}t^2)dt$.

The definition is taken from M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [7.3.2]

Commented Mathematical property (CMP):
S : real $\rightarrow$ real
Commented Mathematical property (CMP):
S : complex $\rightarrow$ complex
Commented Mathematical property (CMP):
S : symbolic $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$S(z)=\sum_{n=0}^{\infty} \frac{(-1)^n(\pi /2)^{2n+1}} {(2n+1)!(4n+3)}z^{4n+3}$, series expansion
Commented Mathematical property (CMP):
$S(-z)=-S(z)$, symmetry
Commented Mathematical property (CMP):
$S(iz)=-iS(z)$, symmetry
Commented Mathematical property (CMP):
$S(\bar{z})=\overline{S(z)}$, symmetry
Signatures:
sts


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LegendreP

The symbol LegendreP represents the Legendre function of the first kind, which is denoted by $P_{\nu}(z)$. It is defined as $P_{\nu}(z)= F(-\nu,\nu+1;1;\frac{1-z}{2})$ where F(...) is the relevant harmonic function, it is a single-valued analytic function in the z-plane excluding z on the real axis from $-\infty$ to $-1$. It satisfies the differential equation $(1-z^2)\frac{d^2w}{dz^2}-2z\frac{dw}{dz}+[\nu(\nu+1) -\frac{\mu^{2}}{1-z^2}]w=0$. The definition is taken from M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [8.1.2]

Commented Mathematical property (CMP):
P : (real, real) $\rightarrow$ real
Commented Mathematical property (CMP):
P : (complex, complex) $\rightarrow$ complex
Commented Mathematical property (CMP):
P : (symbolic, symbolic) $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$P_{\nu}(z)= F(-\nu, \nu+1;1; \frac{1-z}{2})$
Commented Mathematical property (CMP):
$P_{\nu}=P_{\nu}^0$
Signatures:
sts


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LegendreQ

The symbol LegendreQ represents the Legendre function of the second kind, which is denoted by $Q_{\nu}(z)$. It is analytic single-valued function in the z-plane excluding z on the real axis from $-\infty$ to 1.

The definition is taken from M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [8.1.2]

Commented Mathematical property (CMP):
Q : (integer, real) $\rightarrow$ real
Commented Mathematical property (CMP):
Q : (integer, complex) $\rightarrow$ complex
Commented Mathematical property (CMP):
Q : (symbolic, symbolic) $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$Q_{\nu}=Q_{\nu}^0$
Signatures:
sts


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AssociatedLegendreP

The symbol AssociatedLegendreP represents the associated Legendre function of the first kind, which is denoted by $P_{\nu}^{\mu}(z)$ It is defined by $P_{\nu}^{\mu}(z)=\frac{1}{\Gamma(1-\mu)} (\frac{z+1}{z-1})^{\frac{1}{2}\mu}F(-\nu, \nu+1; 1-\mu; \frac{1-z}{2})$ where $F(a, b; c;z)$ is a hypergeometric function and $|1-z|\lt 2$. When $\mu=0$, It reduces to the Legendre function of the first kind $P_{\nu}(z)$. The definition is taken from M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [8.1.2]

Commented Mathematical property (CMP):
P : (real, real, real) $\rightarrow$ real
Commented Mathematical property (CMP):
P : (complex, complex, complex) $\rightarrow$ complex
Commented Mathematical property (CMP):
P : (symbolic, symbolic, symbolic) $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$P_{-\nu-1}^{\mu}(z)=P_{\nu}^{\mu}(z)$
Signatures:
sts


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AssociatedLegendreQ

The symbol AssociatedLegendreQ represents the associated Legendre function of the second kind, which is denoted by $Q_{\nu}^{\mu}(z)$ The associated Legendre function of the second kind is defined by $Q_{\nu}^{\mu}(z)=e^{i\mu\pi}2^{-\nu-1}\sqrt{\pi} \frac{\Gamma(\nu+\mu+1)}{\Gamma(\nu+\frac{3}{2})} z^{-\nu-\mu-1} (z^2-1)^{\frac{1}{2}\mu} F(1+\frac{\nu}{2}+\frac{\mu}{2}, \frac{1}{2}+\frac{\nu}{2} ;\nu+\frac{3}{2}; \frac{1}{2})$ where $|z|>1$. When $\mu=0$, It reduces to the Legendre function of the second kind $Q_{\nu}(z)$. The definition is taken from M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [8.1.3]

Commented Mathematical property (CMP):
Q : (real, real, real) $\rightarrow$ real
Commented Mathematical property (CMP):
Q : (complex, complex, complex) $\rightarrow$ complex
Commented Mathematical property (CMP):
Q : (symbolic, symbolic, symbolic) $\rightarrow$ symbolic
Commented Mathematical property (CMP):
$P_{-\nu-1}^{\mu}(z)=P_{\nu}^{\mu}(z)$
Commented Mathematical property (CMP):
$Q_{-\nu-1}^{\mu}(z)={-\pi e^{i\mu\pi} cos\nu \pi P_{\nu}^{\mu}(z) +Q_{\nu}^{\mu}sin[\pi (\nu+\mu)]}/sin[\pi (\nu-\mu)]$
Signatures:
sts


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