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Why is the orthogonality of complex functions defined with the integration between its complex conjugate?

score 1.7k · Answer 1 · 2016-03-08 00:20:51

1.7k

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Why is the orthogonality of complex functions defined with the integration between its complex conjugate?

Published on 08 Mar 2016 - 0:20

score 171 · Answer 2 · 2016-03-08 02:47:52

171

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Evaluate $\displaystyle \int_{|z-i|=R}\frac{z^{4}+z^{2}+1}{z(z^{2}+1)}dz$ as a function of $R>0$

Published on 08 Mar 2016 - 2:47

score 162 · Answer 3 · 2016-03-08 04:04:16

162

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If $J(L)=\int_{L}\frac{dz}{p(z)}$, where $p(z)$ is a polynomial with $n$ distinct roots, how many values can $J$ take on?

Published on 08 Mar 2016 - 4:04

score 183 · Answer 4 · 2016-03-09 08:08:24

183

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Use Cauchy's Theorem to show that if $\int_{0}^{\infty}f(x)dx$ exists, then so does $\int_{L}f(z)dz$

Published on 09 Mar 2016 - 8:08

score 23 · Answer 5 · 2016-03-09 19:08:22

23

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$\int_{[1+iR, 1+2iR]}\frac{e^{z^2}}{z^2}dz \to 0$ as $R \to \infty$

Published on 09 Mar 2016 - 19:08

score 254 · Answer 6 · 2016-03-10 01:38:29

254

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Use of residues to find I=$\int_0^\infty \frac{\sin^2(x)}{1+x^4} dx$

Published on 10 Mar 2016 - 1:38

score 49 · Answer 7 · 2026-03-28 22:53:43

49

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Converting certain complex exponentials to trigonometric functions

Published on 28 Mar 2026 - 22:53

score 1k · Answer 8 · 2016-03-12 16:57:52

1k

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Determine all the biholomorphic functions $\mathbb{C}\rightarrow \mathbb{C}$.

Published on 12 Mar 2016 - 16:57

score 54 · Answer 9 · 2016-03-13 11:50:16

54

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Calculating complex integral via definition

Published on 13 Mar 2016 - 11:50

score 252 · Answer 10 · 2016-03-14 20:37:35

252

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closed form for $I(n)=\int_0^1\left ( \frac{\pi}{4}-\arctan x \right )^n\frac{1+x}{1-x}\frac{dx}{1+x^2}$

Published on 14 Mar 2016 - 20:37

score 147 · Answer 11 · 2016-03-15 00:17:55

147

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If $f(z)$ analytic on $E(L)$, and $\lim_{z \to \infty}f(z)=A$, prove that $\frac{1}{2\pi i} \int_{L}\frac{f(\zeta)}{\zeta-z}dz = A$ or $-f(z)+A$

Published on 15 Mar 2016 - 0:17

score 264 · Answer 12 · 2016-03-16 17:57:10

264

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Regularity of complex function in region $D$

Published on 16 Mar 2016 - 17:57

score 1.2k · Answer 13 · 2016-03-17 01:16:35

1.2k

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Hermite Polynomials: Rodrigues to Integral Representation

Published on 17 Mar 2016 - 1:16

score 137 · Answer 14 · 2016-03-20 13:13:15

137

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Why $|dz| = -i \rho \frac{dz}{z}$?

Published on 20 Mar 2016 - 13:13

score 62 · Answer 15 · 2026-04-03 14:52:38

62

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Complex integration

Published on 03 Apr 2026 - 14:52

List Question

Why is the orthogonality of complex functions defined with the integration between its complex conjugate?

Evaluate $\displaystyle \int_{|z-i|=R}\frac{z^{4}+z^{2}+1}{z(z^{2}+1)}dz$ as a function of $R>0$

If $J(L)=\int_{L}\frac{dz}{p(z)}$, where $p(z)$ is a polynomial with $n$ distinct roots, how many values can $J$ take on?

Use Cauchy's Theorem to show that if $\int_{0}^{\infty}f(x)dx$ exists, then so does $\int_{L}f(z)dz$

$\int_{[1+iR, 1+2iR]}\frac{e^{z^2}}{z^2}dz \to 0$ as $R \to \infty$

Use of residues to find I=$\int_0^\infty \frac{\sin^2(x)}{1+x^4} dx$

Converting certain complex exponentials to trigonometric functions

Determine all the biholomorphic functions $\mathbb{C}\rightarrow \mathbb{C}$.

Calculating complex integral via definition

closed form for $I(n)=\int_0^1\left ( \frac{\pi}{4}-\arctan x \right )^n\frac{1+x}{1-x}\frac{dx}{1+x^2}$

If $f(z)$ analytic on $E(L)$, and $\lim_{z \to \infty}f(z)=A$, prove that $\frac{1}{2\pi i} \int_{L}\frac{f(\zeta)}{\zeta-z}dz = A$ or $-f(z)+A$

Regularity of complex function in region $D$

Hermite Polynomials: Rodrigues to Integral Representation

Why $|dz| = -i \rho \frac{dz}{z}$?

Complex integration

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