Special Functions¶
Trigonometric functions¶
csc¶
\[ \csc(x) = \frac{1}{\sin(x)} \]
Hyperbolic Functions¶
sinh¶
\[ \sinh z\equiv \frac{1}{2} (e^z-e^{-z}) \]
cosh¶
\[ \cosh z \equiv \frac{1}{2}(e^z+e^{-z}) \]
tanh¶
\[\begin{align}
\tanh (x) &= \frac{\sinh (x)}{\cosh (x)} \\
\frac{\partial \tanh (x)}{\partial x} &= \text{sech}^2(x) \\
\int \tanh (x) \, dx &= \log (\cosh (x))
\end{align}\]
Gamma Function¶
\[\Gamma (z)=\int _0^{\infty } t^{z-1} e^{-t} \,dt\]
Properties & Relations¶
\[\Gamma(z)\Gamma(1-z) \equiv \pi \csc(\pi z)\]
Error Function¶
\[\text{erf} (z)=\frac{2}{\sqrt{\pi }}\int_0^z e^{-t^2} \,dt\]
Complementary Error Function¶
\[ \text{erfc} (z) = 1-\text{erf} (z) \]
Imaginary Error Function¶
\[ \text{erfi} (z) = \frac{\text{erf} (iz)}{i} \]
Bessel Function¶
Bessel function of the first kind¶
\(J_n(z)\) satisfies the differential equation \(z^2y''+zy'+(z^2-n^2)y=0\).
Properties¶
\[\frac{dJ_n(x)}{dx}=\frac{1}{2}(J_{n-1}(x)-J_{n+1}(x))\]
Bessel function of the second kind¶
\(Y_n(z)\) satisfies the differential equation \(z^2y''+zy'+(z^2-n^2)y=0\).
最后更新:
March 8, 2023