Questions tagged [complex-analysis]
For questions mainly about theory of complex analytic/holomorphic functions of one complex variable. Use [tag:complex-numbers] instead for questions about complex numbers. Use [tag:several-complex-variables] instead for questions about holomorphic functions of more than one complex variables.
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To sum $1+2+3+\cdots$ to $-\frac1{12}$
$$\sum_{n=1}^\infty\frac1{n^s}$$
only converges to $\zeta(s)$ if $\text{Re}(s)>1$.
Why should analytically continuing to $\zeta(-1)$ give the right answer?
perplexed
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32
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Evaluating the integral $\int_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2$?
A famous exercise which one encounters while doing Complex Analysis (Residue theory) is to prove that the given integral:
$$\int\limits_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2$$
Well, can ...
user9413
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How to prove Euler's formula: $e^{i\varphi}=\cos(\varphi) +i\sin(\varphi)$?
Could you provide a proof of Euler's formula: $e^{i\varphi}=\cos(\varphi) +i\sin(\varphi)$?
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Proving that a doubly-periodic entire function $f$ is constant.
Let $f: \Bbb C \to \Bbb C$ be an entire (analytic on the whole plane) function such that exists $\omega_1,\omega_2 \in \mathbb{S}^1$, linearly independent over $\Bbb R$ such that: $$f(z+\omega_1)=f(z)=...
- 78.4k
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Nice proofs of $\zeta(4) = \frac{\pi^4}{90}$?
I know some nice ways to prove that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$. For example, see Robin Chapman's list or the answers to the question "Different methods to compute $\sum_{...
- 55.8k
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Frullani 's theorem in a complex context.
It is possible to prove that $$\int_{0}^{\infty}\frac{e^{-ix}-e^{-x}}{x}dx=-i\frac{\pi}{2}$$ and in this case the Frullani's theorem does not hold since, if we consider the function $f(x)=e^{-x}$, we ...
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Show that $\left|\frac{\alpha - \beta}{1-\bar{\alpha}\beta}\right| < 1$ when $|\alpha|,|\beta| < 1$
This is the question I'm stumbling with:
When $|\alpha| < 1$ and $|\beta| < 1$, show that:
$$\left|\cfrac{\alpha - \beta}{1-\bar{\alpha}\beta}\right| < 1$$
The chapter that contains this ...
- 536
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What is a good complex analysis textbook, barring Ahlfors's?
I'm out of college, and trying to self-learn complex analysis. I'm finding Ahlfors' text difficult. Any recommendations? I'm probably at an intermediate sophistication level for an undergrad. (Bonus ...
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Entire function bounded by a polynomial is a polynomial
Suppose that an entire function $f(z)$ satisfies $\left|f(z)\right|\leq k\left|z\right|^n$ for sufficiently large $\left|z\right|$, where $n\in\mathbb{Z^+}$ and $k>0$ is constant. Show that $f$ is ...
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A sine integral $\int_0^{\infty} \left(\frac{\sin x }{x }\right)^n\,\mathrm{d}x$
The following question comes from Some integral with sine post
$$\int_0^{\infty} \left(\frac{\sin x }{x }\right)^n\,\mathrm{d}x$$
but now I'd be curious to know how to deal with it by methods of ...
- 44.4k
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An entire function whose real part is bounded above must be constant.
Greets
This is exercise 15.d chapter 3 of Stein & Shakarchi's "Complex Analysis", they hint: "Use the maximum modulus principle", but I didn't see how to do the exercise with this hint rightaway, ...
- 9,706
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Evaluating $\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx$
How would I go about evaluating this integral?
$$\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx.$$
What I've tried so far: I tried a semicircular integral in the positive imaginary part of the complex ...
user55225
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5
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Definite integral calculation with poles at $0$ and $\pm i\sqrt{3}$
$$\int_0^\infty \frac{\sin(2\pi x)}{x(x^2+3)} \, dx$$
I looked at $\frac{e^{2\pi i z}}{z^{3}+3z}$, also calculated the residues, but they don't get me the right answer. I used that $\int_{-\infty}^\...
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What is wrong with this fake proof $e^i = 1$?
$$e^{i} = e^{i2\pi/2\pi} = (e^{2\pi i})^{1/(2\pi)} = 1^{1/(2\pi )} = 1$$
Obviously, one of my algebraic manipulations is not valid.
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For which complex $a,\,b,\,c$ does $(a^b)^c=a^{bc}$ hold?
Wolfram Mathematica simplifies $(a^b)^c$ to $a^{bc}$ only for positive real $a, b$ and $c$. See W|A output.
I've previously been struggling to understand why does $\dfrac{\log(a^b)}{\log(a)}=b$ and $\...
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