Intereting Posts

Solutions to $p+1=2n^2$ and $p^2+1=2m^2$ in Natural numbers.
Total Time a ball bounces for from a height of 8 feet and rebounds to a height 5/8
Gradient operator the adjoint of (minus) divergence operator?
Computing the sum $\sum_{n=2}^{2011}\sqrt{1+\frac{1}{n^2}+\frac{1}{(n-1)^2}}$
$\operatorname{MaxSpec}(A)$ closed
How is the codomain for a function defined?
Presentation of the fundamental group of a manifold minus some points
Perron-Frobenius theorem
The error term in Taylor series and convolution.
How to solve the equation $3x-4\lfloor x\rfloor=0$ for $x\in\mathbb{R}$?
Dual space and covectors: force, work and energy
Is there a simple proof of the fact that if free groups $F(S)$ and $F(S')$ are isomorphic, then $\operatorname{card}(S)=\operatorname{card}(S')?$
Convergence of $\prod_{n=1}^\infty(1+a_n)$
Evaluate $\int_0^1\int_p^1 \frac {x^3}{\sqrt{1-y^6}} dydx$
Show that $(n!)^{(n-1)!}$ divides $(n!)!$

Let $f\colon\mathbb C \to \mathbb C$ be entire. Show that if

$|\operatorname{Im}f(z)|\leq |\operatorname{Re}f(z)|$ for all $z \in \mathbb C$, then $f$ is constant on $\mathbb C$. How I can answer this by considering the distance between $f(z)$ and $i$.

- Frullani 's theorem in a complex context.
- Euler Product formula for Riemann zeta function proof
- Entire function bounded by polynomial of degree 3/2 must be linear.
- Integral $\int_0^\infty \frac{\sqrt{\sqrt{\alpha^2+x^2}-\alpha}\,\exp\big({-\beta\sqrt{\alpha^2+x^2}\big)}}{\sqrt{\alpha^2+x^2}}\sin (\gamma x)\,dx$
- Getting value of $\sin x + \cos x$ with complex exponential
- If $f$ is an entire function with $|f(z)|\le 100\log|z|$ and $f(i)=2i$, what is $f(1)$?
- Show that the set of zeros of $f$ is discrete
- Find the root of the polynomial?
- The Hairy ball theorem and Möbius transformations
- a generalization of normal distribution to the complex case: complex integral over the real line

$|\mathrm{Im}f(z)|\le |\mathrm{Re}f(z)|$ implies that $|f(z)-i|\ge \dfrac{\sqrt{2}}{2}$. It follows that $g(z)=\dfrac{1}{f(z)-i}$ is a bounded entire function, and due to Liouville’s theorem, it must be a constant.

Alternatively, $\textrm{Re} \, f(z)^2 = \left(\textrm{Re} \, f(z) \right)^2 – \left(\textrm{Im} \, f(z) \right)^2 \geq 0$. This implies that $f(z)^2$ is constant.

follows directly from picard’s little theorem

- Exterior derivative of a complicated differential form
- Counter-example to Cauchy-Peano-Arzela theorem
- Uniqueness of Helmholtz decomposition
- Limit Supremum and Infimum. Struggling the concept
- Geometric Progression: How to solve for $n$ in the following equation $\frac {5^n-1}4 \equiv 2 \pmod 7$
- Motivation behind the definition of Prime Ideal
- Is function invertible?
- We know the dimension of the Koch snowflake's perimeter, but does it have a measure?
- Find the $n$th term of $1, 2, 5, 10, 13, 26, 29, …$
- Why Does SVD Provide the Least Squares and Least Norm Solution to $ A x = b $?
- Showing $f_{n}$ is a normal family, where $f_{n}$ is the $n^{th}$ iterate of $f$
- de Rham comologies of the $n$-torus
- Hahn-Banach Theorem for separable spaces without Zorn's Lemma
- Limit inferior of the quotient of two consecutive primes
- Re-Expressing the Digamma