24758252
| Question | Answer |
| Fourier Series |
f(x)=1√2L∑ncneinπx/L
cn∫L−Lf(x)e−inπx/Ldx |
| Residue Theorem |
∮Cf(z)dz=2πi∑residue
The sum of the residue for singularities contained by the contour.
res(z=z0)=limz→z0[1(p−1)!dp−1dzp−1((z−z0)pf(z))]
where p is the order of the pole.
|
| Symmetric Matrices |
AT=A |
| Antisymmetric Matrices |
AT=−A |
| Orthogonal Matrices |
A−1=AT |
| Hermitian Matrices |
A†=A |
| Antihermitian Matrices |
A†=−A |
| Unitary Matrices |
A−1=A† |
| Normal Matrices |
A†A=AA† |
| Determinant and Trace Based on Eigenvalues |
detA=∏iλi
trA=∑iλi |
| Linear Dependence |
a set is linearly dependent if
n∑i=1ai→vi=0
for any set {ai} except ai=0⋅→vi
→v3=a1→v1+a2→v2 |
| Cauchy-Riemann Conditions |
∂u∂x=∂v∂y
∂v∂x=−∂u∂y |
| Fourier Transform |
Fourier transform of f(x):
˜f(w)=1√2π∫∞−∞f(x)e−iwxdx
Inverse operation:
f(x)=1√2π∫∞−∞˜f(w)eiwxdw |
| Fourier Representation of a Dirac Delta Function |
δ(t)=12π∫∞−∞eiwtdw |
| Pole of Order n |
if n=1 it is a simple pole
limz→z0[(z−z0)nf(z)]≠0
f(z)=g(z)(z−z0)n
g(z) is analytic
|
| Green's Function |
LG(x,x′)=δ(x−x′) |
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