# Tagged Questions

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### Geometric interpretation of complex path integral

Let's say that we want to make sense of integrating a function $f: \mathbb{C}\rightarrow\mathbb{C}$ over some path $\gamma$. I can imagine two reasonable ways of doing it. First, there's the way ...
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### Why are people more interested in the Riemann hypothesis than Goldbach's conjecture? [closed]

One of my friends, a math professor, told me almost every one of his colleagues (in the math department) had attempted to prove the Riemann hypothesis at some point in their life (maybe secretly). ...
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### Path-independent contour integrals and how to define them

If a contour $C$ is parameterized by $z(t): [a, b] \to \mathbb{C}$, then we define $$\int_C f(z) \, dz= \int_a^b f(z(t)) \, z'(t) \, dt.$$ If the contour integral on the left side is equal to some ...
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### Complex analysis is more “real” than real analysis

In physics, in the past, complex numbers were used only to remember or simplify formulas and computations. But after the birth of quantum physics, they found that a thing as real as "matter" itself ...
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### “Geometric” proof of Rouche's theorem on the number of zeros?

I understand the analytic proof of Rouche's theorem as presented in Stein and Shakarchi's complex analysis - $|f(z)| > |g(z)|$ on the boundary circle C ensures that the argument principle can be ...
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### good lecture series for complex analysis?

I was wondering if i could teach myself complex analysis. Any good lecture series out there on the internet available for free (preferably)? One that would be suitable for an undergraduate student?
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### Why study Bergman Spaces?

I'm interested in Operator Algebras and mathematical physics; recently, a friend showed me Duren and Schuster's "Bergman Spaces". I've read about two chapters now and I see there is a nice play ...
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### How to think of zeros of the derivative of a holomorphic funcion?

If $f:\mathbb{R}\rightarrow\mathbb{R}$ is differentiable, then, for example, if $f'(x_0)=0$ and $f'$ is again differentiable at $x_0$ and $f''(x_0)\neq 0$, then $f$ has a maximum or minimum at $x_0$. ...
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### What's the intuition for extending $\mathbb{C}$ to $\mathbb{H}$?

It seems to me that there is a clear, intuitive reason for extending the real number system to the complex number system. Namely, some polynomial equations that have no solutions in $\mathbb{R}$ ...
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### Prove that $\int_{-\pi}^{\pi}$ $\frac{d\theta}{1+\sin^2\theta}$ = $\pi\sqrt{2}$ using the method of Residues

Prove that $$\int_{-\pi}^{\pi}\frac{d\theta}{1+\sin^2\theta} = \pi\sqrt{2}$$ using the method of Residues How do I do this? I know I need it from $0$ to $2\pi$ but I don't know how to modify it!! ...
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### How does one know that a theorem is strong enough to publish?

Question. How does one know that a theorem is strong enough to publish? Basically, I have laid out a framework in which many theorems may be proven. I'm only 18 and therefore lack knowledge of ...