Advanced Topics

 Advanced Topics Solver 

Advanced Topics

161. Fourier Transform: $\widehat{f}(\xi )={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)e^{-2\pi ix\xi }dx$

162. Inverse Fourier Transform: $f(x)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\widehat{f}(\xi )e^{2\pi ix\xi }d\xi$

163. Laplace Transform: $\mathcal{L}\{f(t)\}={\int }_0^{\mathrm{\infty }}f(t)e^{-st}dt$

164. Inverse Laplace Transform: ${\mathcal{L}}^{-1}\{F(s)\}=f(t)$

165. Z-Transform: $X(z)={\sum }_{n=-\mathrm{\infty }}^{\mathrm{\infty }}x[n]z^{-n}$

166. Inverse Z-Transform: $x[n]=\frac{1}{2\pi i}\oint X(z)z^{n-1}dz$

167. Bessel Functions: Solutions to Bessel's differential equation.

168. Legendre Polynomials: Solutions to Legendre's differential equation.

169. Gamma Function: $\mathrm{\Gamma }(z)={\int }_0^{\mathrm{\infty }}{t}^{z-1}{e}^{-t}dt$

170. Beta Function: $B(x,y)={\int }_0^1{t}^{x-1}(1-t{)}^{y-1}dt$

171. Error Function: $\text{erf}(x)=\frac{2}{\sqrt{\pi }}{\int }_0^x{e}^{-t^2}dt$

172. Complementary Error Function: $\text{erfc}(x)=1-\text{erf}(x)$

173. Dirac Delta Function: $\delta (x)$

174. Heaviside Step Function: $H(x)$

175. Green's Function: $G(x,y)$

176. Sturm-Liouville Theory: $\frac{d}{dx}(p(x)\frac{dy}{dx})+q(x)y=\lambda w(x)y$

177. Variational Principle: $\delta \int Ldt=0$

178. Hamiltonian Mechanics: $H=T+V$

179. Lagrangian Mechanics: $L=T-V$

180. Euler-Lagrange Equation: $\frac{\mathrm{\partial }L}{\mathrm{\partial }q}-\frac{d}{dt}\frac{\mathrm{\partial }L}{\mathrm{\partial }\dot{q}}=0$