Flashcards
Mathematical Methods
Flashcards do conjunto (16)
- 1Fourier Series\[f(x)=\frac{1}{\sqrt{2L}}\sum_nc_ne^{in\pi x/L}\]
\[c_n\int_{-L}^{L}f(x)e^{-in\pi x/L}dx\] - 2Residue 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. - 3Symmetric Matrices\[A^T=A\]
- 4Antisymmetric Matrices\[A^T=-A\]
- 5Orthogonal Matrices\[A^{-1}=A^T\]
- 6Hermitian Matrices\[A^{\dagger}=A\]
- 7Antihermitian Matrices\[A^{\dagger}=-A\]
- 8Unitary Matrices\[A^{-1}=A^{\dagger}\]
- 9Normal Matrices\[A^{\dagger}A=AA^{\dagger}\]
- 10Determinant and Trace Based on Eigenvalues\[\text{det}A=\prod_i\lambda_i\]
\[\text{tr}A=\sum_i\lambda_i\] - 11Linear Dependencea set is linearly dependent if
\[\sum_{i=1}^na_i\vec{v}_i=0\]
\[\text{for any set }\{a_i\} \text{ except } a_i=0\cdot\vec{v}_i\]
\[\vec{v}_3=a_1\vec{v}_1+a_2\vec{v}_2\] - 12Cauchy-Riemann Conditions\[\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\]
\[\frac{\partial v}{\partial x}=-\frac{\partial u}{\partial y}\] - 13Fourier TransformFourier transform of f(x):
\[\tilde{f}(w)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(x)e^{-iwx}dx\]
Inverse operation:
\[f(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\tilde{f}(w)e^{iwx}dw\] - 14Fourier Representation of a Dirac Delta Function\[\delta(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{iwt}dw\]
- 15Pole of Order nif \[n=1\] 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 - 16Green's Function\[L\,G(x,x')=\delta(x-x')\]