Question

Show that $\int_{0}^{2\pi}\frac{1}{1+\alpha\cos^2(t)}dt=\frac{2\pi}{\sqrt{\lambda+1}}$ with $\alpha>-1$.

score 55 · Answer 1 · 2026-04-16 04:11:53

55

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Show that $\int_{0}^{2\pi}\frac{1}{1+\alpha\cos^2(t)}dt=\frac{2\pi}{\sqrt{\lambda+1}}$ with $\alpha>-1$.

Published on 16 Apr 2026 - 4:11

score 448 · Answer 2 · 2026-04-16 13:50:43

448

Views

Evaluating the complex integral $\int_{0}^{\infty} \frac {\sin (\ln x) dx }{x^2 + 4} $

Published on 16 Apr 2026 - 13:50

score 66 · Answer 3 · 2026-04-13 16:44:35

66

Views

If f is holomorphic in all $\Bbb{C}$ and $f(z)=f(1-\cos(z))$ $\forall z$, is f necessarily a constant?

Published on 13 Apr 2026 - 16:44

score 924 · Answer 4 · 2026-04-12 04:55:30

924

Views

For a map $F: \mathbb{H} \rightarrow \mathbb{D}$, prove that $|F(z)| \leq |z-i|/|z+i|$ for all $z \in \mathbb{H}$.

Published on 12 Apr 2026 - 4:55

score 90 · Answer 5 · 2026-04-10 14:14:05

90

Views

Maximum principle application

Published on 10 Apr 2026 - 14:14

score 232 · Answer 6 · 2026-04-17 01:51:12

232

Views

Writing $\cos(x^p)$ as a real part of the exponential function

Published on 17 Apr 2026 - 1:51

score 388 · Answer 7 · 2026-04-10 08:21:19

388

Views

Evaluation of a Fresnel type integral.

Published on 10 Apr 2026 - 8:21

score 69 · Answer 8 · 2026-04-09 03:49:38

69

Views

Can you use complex intervals for definite integrals?

Published on 09 Apr 2026 - 3:49

score 122 · Answer 9 · 2026-04-11 23:07:01

122

Views

Ahlfors' Differentiation

Published on 11 Apr 2026 - 23:07

score 942 · Answer 10 · 2026-04-14 15:41:52

942

Views

Conformal map from $\mathbb C\setminus \{0\}$ to the open unit disk.

Published on 14 Apr 2026 - 15:41

score 36 · Answer 11 · 2026-04-11 19:07:29

36

Views

Proof: $f(z)=Ce^{az}$

Published on 11 Apr 2026 - 19:07

score 1.3k · Answer 12 · 2026-04-28 15:26:20

1.3k

Views

Why is a line integral of a conservative vector field independent of path?

Published on 28 Apr 2026 - 15:26

score 54 · Answer 13 · 2026-04-09 10:06:41

54

Views

holomorophic function on unit disk

Published on 09 Apr 2026 - 10:06

score 197 · Answer 14 · 2026-04-12 12:35:27

197

Views

Analytic continuation on disconnected sets

Published on 12 Apr 2026 - 12:35

score 245 · Answer 15 · 2026-04-15 22:31:46

245

Views

Prove that $(1 - 2^{1-z})\zeta(z) = 1^{-z} - 2^{-z} + 3^{-z} - \cdots$

Published on 15 Apr 2026 - 22:31

List Question

Show that $\int_{0}^{2\pi}\frac{1}{1+\alpha\cos^2(t)}dt=\frac{2\pi}{\sqrt{\lambda+1}}$ with $\alpha>-1$.

Evaluating the complex integral $\int_{0}^{\infty} \frac {\sin (\ln x) dx }{x^2 + 4} $

If f is holomorphic in all $\Bbb{C}$ and $f(z)=f(1-\cos(z))$ $\forall z$, is f necessarily a constant?

For a map $F: \mathbb{H} \rightarrow \mathbb{D}$, prove that $|F(z)| \leq |z-i|/|z+i|$ for all $z \in \mathbb{H}$.

Maximum principle application

Writing $\cos(x^p)$ as a real part of the exponential function

Evaluation of a Fresnel type integral.

Can you use complex intervals for definite integrals?

Ahlfors' Differentiation

Conformal map from $\mathbb C\setminus \{0\}$ to the open unit disk.

Proof: $f(z)=Ce^{az}$

Why is a line integral of a conservative vector field independent of path?

holomorophic function on unit disk

Analytic continuation on disconnected sets

Prove that $(1 - 2^{1-z})\zeta(z) = 1^{-z} - 2^{-z} + 3^{-z} - \cdots$

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