Clausen(x, min_tol=1e-15)

Computes the Clausen function

\[ Cl_2(\phi) = - \int_0^\phi \log|2\sin(x/2)| dx\]

Returns $Cl_2(\phi)$.

Debye_function(n,x,min_tol=1e-15)

The Debye function(n,x) given by

\[ D_n(x) = \frac{n}{x^n} \int_0^x \frac{t^n}{e^{t}-1} dt\]

Returns the value $D(n,x)$

regular_coulomb(ℓ,η,ρ)

Regular Coulomb wave function ℓ is the order(non-negative integer), η is the charge (real parameter) and ρ is the radial coordinate (non-negative real variable).

returns the value F_ℓ(η,ρ) given by

\[ F_\ell(\eta,\rho) = \frac{\rho^{\ell+1}2^\ell e^{i\rho-(\pi\eta/2)}}{|\Gamma(\ell+1+i\eta)|} \int_0^1 e^{-2i\rho t}t^{\ell+i\eta}(1-t)^{\ell-i\eta} \, dt\]

irregular_Coulomb(ℓ,η,ρ)

Regular Coulomb wave function ℓ is the order(non-negative integer), η is the charge (real parameter) and ρ is the radial coordinate (non-negative real variable).

returns the value G_ℓ(η,ρ)

C(ℓ,η)

Returns Coulomb normalization constant given by

\[ C_\ell(\eta) = \frac{2^\ell \exp(-\pi \eta/2) |\Gamma(\ell+1+i \eta)|}{(2\ell+1)!}\]

θ(ℓ,η,ρ)

Returns the phase of the Coulomb functions given by

\[ \theta_\ell(\eta,\rho) = \rho - \eta \ln(2\rho) - \frac{1}{2}\ell \pi + \sigma_\ell(\eta)\]

Coulomb_H_minus(ℓ,η,ρ)

Complex Coulomb wave function. Infinity handled using the substitution f(t) -> f(u/(1-u)*1/(1-u)^2). Returns Coulomb wave function

\[ H^{-}_\ell = G_\ell - iF_\ell\]

Returns Coulomb wave function

\[ H^{+}_\ell = G_\ell + iF_\ell\]

Coulomb_cross(ℓ,η)

Wronskian relation / cross product.

\[ F_{\ell-1}G_{\ell}-F_{\ell}G_{\ell-1} = \ell/(\ell^2+\eta^2)^{1/2}\]

regular_Coulomb_approx(ℓ,η,ρ)

For ρ -> 0 and η fixed approximate the regular Coulomb wave function as

\[ F_\ell(\eta,\rho) \simeq C_\ell(\eta)^{\ell+1}\]

irregular_Coulomb_approx(ℓ,η,ρ)

For ρ -> 0 and η fixed approximate the irregular Coulomb wave function as

\[ G_\ell(\eta,\rho) \simeq \frac{\rho^{-\ell}}{(2\ell+1)C_\ell(\eta)}\]

regular_Coulomb_limit(ℓ,η,ρ)

In the limit ρ -> ∞ with η fixed, returns the regular Coulomb wave as

\[ F_{\ell}(\eta,\rho) \simeq \sin(\theta_\ell(\eta,\rho))\]

irregular_Coulomb_limit(ℓ,η,ρ)

In the limit ρ -> ∞ with η fixed, returns the irregular Coulomb wave as

\[ G_{\ell}(\eta,\rho) \simeq \cos(\theta_\ell(\eta,\rho))\]

Struve(ν,z,min_tol=1e-15)

Returns the Struve function given by

\[ \mathbf{H}_\nu(z) = \frac{2(z/2)^\nu}{\sqrt{\pi}\Gamma(\nu+1/2)} \int_0^1 (1-t)^{{\nu-1/2}}\sin(zt) \, \text{d}t\]

Fresnel_S_integral_pi(x)

The Fresnel function S(z) using the definition in Handbook of Mathematical Functions: Abramowitz and Stegun, where

\[ S(z) = \int_0^x \cos(\pi t^2/2) dt\]

Returns the value $S(x)$

Fresnel_C_integral_pi(x)

The Fresnel function C(z) using the definition in Handbook of Mathematical Functions: Abramowitz and Stegun, where

\[ C(z) = \int_0^x \sin(\pi t^2/2) dt\]

Returns the value $C(x)$

Fresnel_S_integral(x)

The Fresnel function S(z) using the definition wiki

\[ S(z) = \int_0^x \sin(t^2) dt\]

Returns the value $S(x)$

Fresnel_C_integral(x)

The Fresnel function C(z) using the definition wiki

\[ C(z) = \int_0^x \cos(t^2) dt\]

Returns the value $C(x)$

Fresnel_S_erf(x)

The Fresnel function S(z) using the definition wiki and the error function.

\[ S(z) = \sqrt{\frac{\pi}{2}}\frac{1+i}{4} \bigg( \text{erf}\big(\frac{1+i}{\sqrt{2}}z \big) - i \text{erf}\big(\frac{1-i}{\sqrt{2}}z \big)\bigg)\]

Returns the value $S(x)$

Fresnel_C_erf(x)

The Fresnel function C(z) using the definition wiki and the error function.

\[ C(z) = \sqrt{\frac{\pi}{2}}\frac{1-i}{4} \bigg( \text{erf}\big(\frac{1+i}{\sqrt{2}}z \big) + i \text{erf}\big(\frac{1-i}{\sqrt{2}}z \big)\bigg)\]

Returns the value $C(x)$

hypergeometric_0F1(b,z)

Returns the confluent hypergeometric function given by

\[ {}_0 F_1(a,b) = \sum_{k=0}^\infty \frac{z^k}{(b)_k k!}\]

for the parameters $a$ and $b$

confluent_hypergeometric_1F1(a,b,z)

Returns the Kummer confluent hypergeometric function

\[ {}_1 F_1(a,b,z) = \sum_{k=0}^{\infty} \frac{(a)_k z^k}{(b)_k k!}\]

confluent_hypergeometric_U(a,b,z)

Returns the Kummer confluent hypergeometric function

\[ U(a,b,z) = \frac{\Gamma(b-1)}{\Gamma(a)}z^{1-b} {}_1 F_1(a-b+1,2-b,z)+\frac{\Gamma(1-b)}{\Gamma(a-b+1)} {}_1F_1(a,b,z)\]

FermiDiracIntegral(j,x)

The Fermi-Dirac integral

\[ F_j(x) = \int_0^\infty \frac{t^j}{\exp(t-x)+1} \, dt\]

Returns the value $F_j(x)$

Resources: [1] D. Bednarczyk and J. Bednarczyk, Phys. Lett. A, 64, 409 (1978) [2] J. S. Blakemore, Solid-St. Electron, 25, 1067 (1982) [3] X. Aymerich-Humet, F. Serra-Mestres, and J. Millan, Solid-St. Electron, 24, 981 (1981) [4] X. Aymerich-Humet, F. Serra-Mestres, and J. Millan, J. Appl. Phys., 54, 2850 (1983) [5] H. M. Antia, Rational Function Approximations for Fermi-Dirac Integrals (1993)

https://arxiv.org/abs/0811.0116 https://de.wikipedia.org/wiki/Fermi-Dirac-Integral https://dlmf.nist.gov/25.12#iii

FermiDiracIntegralNorm(j,x)

The Fermi-Dirac integral

\[ F_j(x) = \frac{1}{\Gamma(j+1)}\int_0^\infty \frac{t^j}{\exp(t-x)+1} \, dt\]

Returns the value $F_j(x)$

Resources: [1] D. Bednarczyk and J. Bednarczyk, Phys. Lett. A, 64, 409 (1978) [2] J. S. Blakemore, Solid-St. Electron, 25, 1067 (1982) [3] X. Aymerich-Humet, F. Serra-Mestres, and J. Millan, Solid-St. Electron, 24, 981 (1981) [4] X. Aymerich-Humet, F. Serra-Mestres, and J. Millan, J. Appl. Phys., 54, 2850 (1983) [5] H. M. Antia, Rational Function Approximations for Fermi-Dirac Integrals (1993)

https://arxiv.org/abs/0811.0116 https://de.wikipedia.org/wiki/Fermi-Dirac-Integral https://dlmf.nist.gov/25.12#iii