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Serafina
  • Member for 4 years, 8 months
  • Last seen more than a month ago
5 votes
3 answers
219 views

Trying to integrate $\int_0^{+\infty} \frac{x^{1/3}\ln x}{x^2+4}\; dx$

5 votes
1 answer
240 views

Suppose $f$ is an entire function such that $f(0) = 1$ and $\int_0^{2\pi} |f(e^{i\theta})| d\theta = 2\pi$. Show that $f$ must be a constant function.

4 votes
2 answers
197 views

Evaluate the definite integral $I = \int_0^\pi \frac{\sin^2(\theta) d\theta}{10-6\cos(\theta)}$

4 votes
0 answers
207 views

Find all analytic bijections $f: \mathbb{C} \to \mathbb{C}$. Justify that there are no other analytic bijections besides those you found.

4 votes
1 answer
198 views

For $u$ harmonic and $f = u+iv$ holomorphic, show that $f(z) = \frac{1}{\pi i} \oint_{|\zeta|=r} \frac{u(\zeta)}{\zeta - z} d\zeta - \overline{f(0)}$

3 votes
0 answers
81 views

Let $z, w$ be complex numbers lying in the first quadrant. Prove that $|z|^2 + |w|^2 \leq |z + w|^2 \leq 2(|z|^2 + |w|^2)$.

2 votes
3 answers
213 views

Taylor expansion and coefficients of $f(z) = \frac{z}{e^z-1}$.

2 votes
2 answers
124 views

Expand the function $f(z) = \frac{1+2z^2}{z^2+z^4} $ into power series of $z$ in all areas of convergence.

2 votes
3 answers
129 views

Evaluate $\underset{z=0}{\text{Res}} \; \frac{(z^6-1)^2}{z^5(2z^4 -5z^2 + 2)}$ .

2 votes
2 answers
169 views

Contour Integral around the unit circle $C$: $\oint_C \frac{e^z-1}{\sin^3(z)}dz$

1 vote
0 answers
84 views

Uniform convergence of a sequence of holomorphic functions on the unit circle and disc

1 vote
1 answer
296 views

Determine the largest possible value of $|f(1/2)|$ and give an example of a function $f$ that attains your upper bound.

1 vote
1 answer
529 views

If $f(z)$ is a 1-to-1 analytic function on the unit disk and $f(0)=0$, show that there is an analytic function $g(z)$ such that $g(z)^2 = f(z^2)$.

1 vote
0 answers
122 views

Consider the function $f(z) = \frac{z+6}{z^2-2z-3} $. (Taylor Series and Laurent Series problem)

1 vote
1 answer
177 views

Suppose $f(z)$ is analytic on $\mathbb{C}\setminus\{0\}$ and $|f(z)| \leq |z|^{5/2} + |z|^{-1/2}$ for all $z\in \mathbb{C}\setminus\{0\}$...

1 vote
2 answers
2k views

Suppose $f(z)$ is analytic on $|z|<1$ such that $|f(z)|<1$ for all $|z|<1$ and $f(0) = \alpha \neq 0$. Show $f(z) \neq 0$ for all $|z|<|\alpha|$. [duplicate]

1 vote
1 answer
610 views

Suppose $f$ is holomorphic on $\mathbb{C}\setminus\{0\}$ and $f(n)=(-1)^n$ for each positive integer $n$. Prove $\inf_{z\neq 0} |f(z)|=0$.

0 votes
1 answer
54 views

Let $A$ be the set of points where the gradient of $u$ vanishes. If $A$ has an accumulation point in $\Omega$, is it true that $u$ is constant?

0 votes
1 answer
85 views

$f(z)$ analytic on $D$ with sequence $z_n \in D$ s.t. $|f(z_n)| \to \infty$ as $n \to \infty$. Radius of convergence of power series for $f$ is 1?

0 votes
1 answer
117 views

Determine if analytic mappings from $\mathbb{C}\setminus \{0\}$ and $\mathbb{C}\setminus [0,\infty)$ to open unit disk exist and if so, find them...

0 votes
2 answers
150 views

Find how many solutions (counting multiplicity) the equation $\sin z = ez^4$ has on the unit disk $|z|<1$.

0 votes
1 answer
69 views

Verify that $u, \; v$ are continuous in a neighborhood of $z=0$ and satisfy the Cauchy-Riemann Eqns at $z=0$. Show that $f'(0)$ does not exist.

0 votes
1 answer
213 views

Show that $\cos(\sqrt{z}) = z$ has infinitely many solutions in $\mathbb{C}$.