# Tagged Questions

Questions on the evaluation of integrals along a locus in the complex plane.

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### Cauchy's Integral Formula - clarification on permissible closed curves

The Cauchy Integral Formula that I am working with says: Suppose that $f:E \rightarrow \mathbb{C}$ is holomorphic, $E$ is an open subset of $\mathbb{C}$, and $z_0 \in E$. Pick $\rho > 0$ such ...
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### Does the analytical form of the following integral exist?

I have an integral $$\int_0^{2\pi}d\theta\cos(2\theta)e^{-a[1-\cos(\theta-\theta_0)]}.$$ Is there any analytical form for the integral above?
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### What is meant by the “contour of a function?”

Suppose that we have $f(x,y)=(x+y)^2.$ What is meant by the "contour of a function," and what is an analytic expression for it? All software, such as Matlab, Mathematica,.. gives just a function like ...
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### Use a rectangular contour to evaluate the integral

$$\int_{-\infty}^{\infty} \frac{\cos(mx) dx}{e^{-x}+e^x} = \frac{\pi}{e^{m\pi /2}+e^{-m\pi /2}}$$ I need to evaluate the above integral specifically using a rectangluar contour and at some point ...
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### Piecewise continuous contours with discontinuity only at end points

Let $w(t)=u(t)+iv(t)$ where $a \leq t \leq b$ be a complex valued function on real variable $t$. For integrating $w(t)$ from $a$ to $b$ we require that $u(t)$ and $v(t)$ must be piecewise continuous....
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### What is Complex Analysis? Why is it accompanied by Linear Algebra?

I hope this doesn't extend to a lengthy question. I studied Linear Algebra recently in my first term at university. I came to the realization however that some institutions would teach that course ...
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### Evaluate using complex integration: $\int_{-\infty}^\infty \frac{dx}{(x^2+1)(x^2+9)}$

Evaluate $$\int_{-\infty}^\infty \frac{dx}{(x^2+1)(x^2+9)}$$ Firsly I found the residues of this function: $$Res(i)=-i/16$$ $$Res(-i)=i/16$$ $$Res(3i)=i/48$$ $$Res(-3i)=-i/48$$ I then closed ...
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