24758252
| Question | Answer |
| Fourier Series | \[f(x)=\frac{1}{\sqrt{2L}}\sum_nc_ne^{in\pi x/L}\] \[c_n\int_{-L}^{L}f(x)e^{-in\pi x/L}dx\] |
| Residue Theorem | \[\oint_Cf(z)dz=2\pi i\sum\text{residue}\] The sum of the residue for singularities contained by the contour. \[\text{res}(z=z_0)=\lim{z\to z_0}\Big[\frac{1}{(p-1)!}\frac{d^{p-1}}{dz^{p-1}}((z-z_0)^pf(z))\Big]\] where p is the order of the pole. |
| Symmetric Matrices | \[A^T=A\] |
| Antisymmetric Matrices | \[A^T=-A\] |
| Orthogonal Matrices | \[A^{-1}=A^T\] |
| Hermitian Matrices | \[A^{\dagger}=A\] |
| Antihermitian Matrices | \[A^{\dagger}=-A\] |
| Unitary Matrices | \[A^{-1}=A^{\dagger}\] |
| Normal Matrices | \[A^{\dagger}A=AA^{\dagger}\] |
| Determinant and Trace Based on Eigenvalues | \[\text{det}A=\prod_i\lambda_i\] \[\text{tr}A=\sum_i\lambda_i\] |
| Linear Dependence | a set is linearly dependent if \[\sum_{i=1}^na_i\vec{v}_i=0\] \[\text{for any set }\{a_i\} \text{ except } a_i=0\cdot\vec{v}_i\] \[\vec{v}_3=a_1\vec{v}_1+a_2\vec{v}_2\] |
| Cauchy-Riemann Conditions | \[\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\] \[\frac{\partial v}{\partial x}=-\frac{\partial u}{\partial y}\] |
| Fourier Transform | Fourier transform of f(x): \[\tilde{f}(w)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(x)e^{-iwx}dx\] Inverse operation: \[f(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\tilde{f}(w)e^{iwx}dw\] |
| Fourier Representation of a Dirac Delta Function | \[\delta(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{iwt}dw\] |
| Pole of Order n | if \[n=1\] it is a simple pole \[\lim{z\to z_0}\Big[(z-z_0)^nf(z)\Big]\neq0\] \[f(z)=\frac{g(z)}{(z-z_0)^n}\] g(z) is analytic |
| Green's Function | \[L\,G(x,x')=\delta(x-x')\] |
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