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<math>Ai(x) = (1/\pi) \int_0^\infty \cos((1/3) t^3 + xt) dt</math></br> |
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<math>Ai(x) = (1/\pi) \int_0^\infty \cos((1/3) t^3 + xt) dt</math></br> |
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<math>Bi(x) = (1/\pi) \int_0^\infty (e^(-(1/3) t^3) + \sin((1/3) t^3 + xt)) dt</math> |
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<math>Bi(x) = (1/\pi) \int_0^\infty (e^(-(1/3) t^3) + \sin((1/3) t^3 + xt)) dt</math> |
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== Clausen Functions == |
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<math>Cl_2(x) = - \int_0^x dt \log(2 \sin(t/2))</math> |
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== Normalized Hydrogenic Bound States == |
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<math>R_n := 2 (Z^{3/2}/n^2) \sqrt{(n-l-1)!/(n+l)!} \exp(-Z r/n) (2Zr/n)^l L^{2l+1}_{n-l-1}(2Zr/n)</math></br> |
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Where:<math>L^k_n(x) = (-1)^k (d^k/dx^k) L_(n+k)(x)</math> |
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== Legendre Forms == |
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<math> F(\phi,k) = \int_0^\phi dt 1/\sqrt((1 - k^2 \sin^2(t)))</math> |
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<math> E(\phi,k) = \int_0^\phi dt \sqrt((1 - k^2 \sin^2(t)))</math> |
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<math> \Pi(\phi,k,n) = \int_0^\phi dt 1/((1 + n \sin^2(t))\sqrt(1 - k^2 \sin^2(t))</math> |
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== Carlson Forms == |
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<math>RC(x,y) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1)</math> |
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<math> RD(x,y,z) = 3/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-3/2)</math> |
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<math> RF(x,y,z) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2)</math> |
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<math> RJ(x,y,z,p) = 3/2 \int_0^\infty dt(t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2) (t+p)^(-1)</math> |
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== Gamma Functions == |
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<math> \Gamma(x) = \int_0^\infty dt t^{x-1} \exp(-t)</math> |
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== Psi (Digamma) Function == |
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<math> \psi^{(n)}(x) = (d/dx)^n \psi(x) = (d/dx)^{n+1} \log(\Gamma(x)) </math> |
Akanksh Vashisth
Complete Fermi Dirac Integrals
Incomplete Fermi Dirac Integrals
Airy Functions and Derivatives
Clausen Functions
Normalized Hydrogenic Bound States
Where:
Gamma Functions
Psi (Digamma) Function