Intereting Posts

If $(a + ib)^3 = 8$, prove that $a^2 + b^2 = 4$
What is the difference between writing f and f(x)?
$f \in L^1$, but $f \not\in L^p$ for all $p > 1$
A convex function is differentiable at all but countably many points
Sum of GCD(k,n)
Is it possible to formalize all mathematics in terms of ordinals only?
Approximation of Shannon entropy by trigonometric functions
$S=\{(n,{1\over n}):n\in\mathbb{N}\}$ is closed in $X$?
Show that the number or $r$ combinations of $X$ which contain no consecutive integers is given by $\binom{n-r+1}{r}$
Simple linear map question
Lower bound for finding second largest element
Equivalence of following statements about shortest path problem
Show that $\sum_{n \le x} \phi (n)=\frac{x^2}{2\zeta(2)}+ O(x \log x)$
Example of prime, not maximal ideal
Are there “essentially non-constructive” statements?

Plain and simple: Can each entire function $f$ without zeros be written in the form $e^g$ for some entire function $g$? What would be a counter-example?

- Questions aobut Weierstrass's elliptic functions
- Field Extension problem beyond $\mathbb C$
- When can't a real definite integral be evaluated using contour integration?
- Explicitly reconstructing a function from its moments
- holomorphic functions and fixed points
- Finding the Laurent series of $f(z)=\frac{1}{(z-1)^2}+\frac{1}{z-2}$?
- entire bijection of $\mathbb{C}$ with 2 fixed points
- How to find Laurent series Expansion
- Show $\int_0^\infty \frac{\cos a x-\cos b x}{\sinh \beta x}\frac{dx}{x}=\log\big( \frac{\cosh \frac{b\pi}{2 \beta}}{\cosh \frac{a\pi}{2\beta}}\big)$
- Continuity of analytic function implies convergence of power series?

Every zero-free holomorphic function $f$ on a simply connected domain $\Omega \subset \mathbb{C}$ has a logarithm.

Since $\Omega$ is simply connected, the logarithmic derivative of $f$ has a primitive $h$, i.e.

$$h'(z) = \frac{f'(z)}{f(z)}.$$

That means the function $f\cdot e^{-h}$ has derivative $\equiv 0$, so is constant. Choosing an appropriate constant, we obtain

$$f(z) = e^{h(z) + c}.$$

Since $\mathbb{C}$ is simply connected, that applies in particular to entire functions.

- Explanation Borel set
- Show $\int_{0}^{\frac{\pi}{2}}\frac{x^{2}}{x^{2}+\ln^{2}(2\cos(x))}dx=\frac{\pi}{8}\left(1-\gamma+\ln(2\pi)\right)$
- cup product in cohomology ring of a suspension
- GCD in a PID persists in extension domains
- Factoring inequalities on Double Summation (Donald Knuth's Concrete Mathematics)
- Different methods of evaluating $\int\sqrt{a^2-x^2}dx$:
- Find all such functions defined on the space
- continuous map on $\mathbb{R}$ which is the identity on $\mathbb{Q}$ is the identity map, hence Aut$(\mathbb{R}/\mathbb{Q})= 1.$
- Prove that $ f:(a,b)\to\mathbb{R}$ is integrable iff $\lim_{\epsilon\to0} \int_{}f$ exists
- Does Euclid lemma hold for GCD domains?
- What is an odd function?
- Finite Group and normal Subgroup
- Problems with Inequalities
- Proving Nicomachus's theorem without induction
- The rank of a linear transformation/matrix