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Apr
28
accepted Suppose $u,v \in \mathbb{C}$ are in the open unit disk. Is $|u|^n - |v|^n \leq |u - v|^n$?
Apr
16
accepted Let $z=x+iy$. What is $\frac{\partial}{\partial x} \log|z-w|$ and $\frac{\partial}{\partial y} \log|z-w|$?
Apr
4
accepted All solutions of $A^2+I=0$ in $M(n,\mathbb{C})$ are similar
Apr
4
accepted Let $V$ be the space of complex polynomials on $[0,1]$. Is the differentiation operator self-adjoint?
Mar
26
accepted What does it mean for a function to tend uniformly to $\infty$ on every compact set?
Feb
23
accepted Let $f(z) = z^4 - 2z^3 + z^2$. Evaluate $\frac{1}{2\pi i} \int \frac{f'}{f} dz$ and $\int \frac{zf'}{f} dz$
Feb
17
accepted What is the form of the general mobius transformations that map the line $\Re(z)=2$ and the unit circle into concentric circles?
Feb
9
accepted What general mobius transformation maps $|z-1|=1$ to itself and $|z+1|=1$ to $|w-3|=3$.
Feb
3
accepted If $f$ is analytic in $D$ and $|f(z)|<M$ everywhere on $|z|=1$, show for all $z:|z|<1$, $|f(z)| \leq M |\frac{z-a}{\bar a z - 1}|$
Feb
1
accepted Suppose $T$ is diagonalizable in $\mathbb{C}$. Show $e^T = \sum_{\lambda \in sp(T)} e^\lambda P_\lambda$ is the matrix exponential series.
Jan
18
accepted (Ahlfors, p198) Why is it clear we can write $G(z-1)=ze^{\gamma(z)}G(z)$ when deriving the Gamma function?
Jan
17
accepted Laurent expansion of $1/(1+z^n)$ for $n \in \mathbb{N}$.
Jan
13
accepted Let $f(x) = (x^n-1)/(x-1)$. Why does $f(1)=n$?
Jan
2
accepted Let $f(z) = \frac{z^{-2}}{\sin( \pi z )}$. What is the residue for $z \neq 0$?
Dec
19
accepted What is the statistical steady state of this poisson process?
Dec
18
accepted Suppose $dX_t = a(X_t) dt + b(X_t) dW_t$ and $Y_s=X_t$ where $s=t^2$. What SDE does $Y_s$ satisfy in the weak sense?
Dec
13
accepted Suppose $X_t$ is a brownian motion with $X_0 \sim u_0$. What is the probability density of $X_t$? (heat equation)
Nov
20
accepted How to solve the following PDE for A and B?
Nov
4
accepted If $T^2=O$ where $O$ is $O(\vec{x})=\vec{0}$, prove that $T$ has no inverse
Nov
4
accepted If $A$ is a rank one linear transformation, show there is a unique scalar $\alpha$ such that $A^2 = \alpha A$