Intereting Posts

Existence of increasing sequence $\{x_n\}\subset S$ with $\lim_{n\to\infty}x_n=\sup S$
For which $d \in \mathbb{Z}$ is $\mathbb{Z}$ a unique factorization domain?
Prove or the disprove the existence of a limit of integrals
Proof by Strong Induction for $a_k = 2~a_{k-1} + 3~a_{k-2}$
What is the least amount of questions to find out the number that a person is thinking between 1 to 1000 when they are allowed to lie at most once
Dividing the linear congruence equations
Preparing for Spivak
Unpacking the Diagonal Lemma
A ring isomorphic to its finite polynomial rings but not to its infinite one.
Construct a complete metric on $(0,1)$
$P_n(x):=1+ \sum_{m=1}^n\dfrac{x^m}{m!}$ has no real root for even $n$ and exactly one real root for odd $n$
A very general method for proving inequalities. Too good to be true?
Entropy of matrix
Star convex set is simply connected.
Eigenvalues of block matrix related

Prove if $f$ and $g$ are entire and $e^f+e^g=1$, then $f$ and $g$ are constant.

I believe the simplest way would be to use Louiville’s theorem by using Pick’s theorem but I am not sure on how to go about this.

- Entire, $|f(z)|\le1+\sqrt{|z|}$ implies $f$ is constant
- Control on Conformal map
- Evaluating real integral using residue calculus: why different results?
- Removable singularity and laurent series
- Projections on the Riemann Sphere are antipodal
- Show that $\int\nolimits^{\infty}_{0} x^{-1} \sin x dx = \frac\pi2$

- If there is a branch of $\sqrt{z}$ on an open set $U$ with $0 \notin U,$ then there is also a branch of $arg$ $z.$
- Let $f$ be entire and suppose that $\text{Im} f\ge0$. Show that $\text{Im} f$ is constant.
- Harmonic Function with linear growth
- Can the “radius of analyticity” of a smooth real function be smaller than the radius of convergence of its Taylor series without being zero?
- Residues and poles proof
- Simplification of $G_{2,4}^{4,2}\left(\frac18,\frac12\middle|\begin{array}{c}\frac12,\frac12\\0,0,\frac12,\frac12\\\end{array}\right)$
- Uniform convergence of the Bergman kernel's orthonormal basis representation on compact subsets
- Integral $\int_0^\infty \frac{\sin^2 ax}{x(1-e^x)}dx=\frac{1}{4}\log\left( \frac{2a\pi}{\sinh 2a\pi}\right)$
- Calculating the integral $\int_0^{\infty}{\frac{\ln x}{1+x^n}}$ using complex analysis
- Show that an entire function $f$ s.t. $|f(z)|>1$ for $|z|>1$ is a polynomial

Use the little Picard theorem:

If a function $f : \mathbb C\rightarrow \mathbb C$ is entire and non-constant, then the set of values that $f(z)$ assumes is either the whole complex plane or the plane minus a single point.

The range of $e^f$ doesn’t contain $0$. It also doesn’t contain $1$ (otherwise we would have: $1 + e^g = 1 \implies e^g = 0$, a contradiction). Hence, it’s constant.

- Tower property of conditional expectation
- What are some examples of theories stronger than Presburger Arithmetic but weaker than Peano Arithmetic?
- The Duals of $l^\infty$ and $L^{\infty}$
- To Find The Exponential Of a Matrix
- To prove $ \tan(A) + 2 \tan(2A) +4\tan4A + 8 \cot8A =\cot(A) $
- How does handle attachment work in Morse Theory
- how to prove $m^{\phi(n)}+n^{\phi(m)}\equiv 1 \pmod{mn}$ where m and n are relatively prime?
- If a field $F$ is such that $\left|F\right|>n-1$ why is $V$ a vector space over $F$ not equal to the union of proper subspaces of $V$
- positive martingale process
- Ramanujan's transformation formula connected with $r_{2}(n)$
- gradient descent optimal step size
- Hamel basis for $\mathbb{R}$ over $\mathbb{Q}$ cannot be closed under scalar multiplication by $a \ne 0,1$
- Do the Kolmogorov's axioms permit speaking of frequencies of occurence in any meaningful sense?
- Compound Distribution — Log Normal Distribution with Log Normally Distributed Mean
- Solving equations of form $3^n – 1 \bmod{k} = 0$, $k$ prime