Intereting Posts

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Finding the first digit of $2015^{2015}$
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Is a matrix multiplied with its transpose something special?

There are many theorems in complex analysis which tell us about integration $\int_{\gamma} f$ where $f$ is continuous (or even differentiable) in the interior of $\gamma$ except finitely many points.

I would like to see how should we solve $\int_{\gamma}f(z)dz$ where $\gamma$ contains a singular point of $f$.

For example, in solving $\int_{|z|=1} \frac{1}{z-1}dz$, I slightly pulled the curve $|z|=1$ near $1$ towards left, so that $\gamma$ will not contain $1$; then the singularity of $\frac{1}{z-1}$ will not be inside this deformed curve, so integration over deformed curve is zero, so taking limit, I concluded that $\int_{|z|=1} \frac{1}{z-1}dz=0$.

- Perron's formula (Passing a limit under the integral)
- Why can't an analytic function be injective in a neighborhood of an essential singularity?
- Using the identity theorem: can there be an analytic function $f$ with $f\left(\frac{1}{n^2}\right) = \frac{1}{n}$
- Integral $\int_0^{\pi/2} \frac{\sin^3 x\log \sin x}{\sqrt{1+\sin^2 x}}dx=\frac{\ln 2 -1}{4}$
- Using Residue theorem to evaluate $ \int_0^\pi \sin^{2n}\theta\, d\theta $
- Cauchy-Riemann equations in polar form.

However, if I pull the curve near $1$ on the right side, then the singularity of $\frac{1}{z-1}$ will be inside this deformed curve, and integration is then non-zero.

I confused between these two processes; can you help me what is the correct way to proceed for finding integration along curves, in which singularity of the function is on the curve?

- Integral $\int_0^\infty \log^2 x\frac{1+x^2}{1+x^4}dx=\frac{3 \pi^3}{16\sqrt 2}$
- Relationship Between Ratio Test and Power Series Radius of Convergence
- Prove that $f(z)=\frac{1}{2\pi}\int_0^{2\pi} f(Re^{i\phi})Re(\frac{Re^{i\phi}+z}{Re^{i\phi}-z}) d\phi$
- Does a convergent power series on a closed disk always converge uniformly?
- Singular points
- Zeros of analytic function and limit points at boundary
- Prove that $\prod_{n=2}^∞ \left( 1 - \frac{1}{n^4} \right) = \frac{e^π - e^{-π}}{8π}$
- how to show image of a non constant entire function is dense in $\mathbb{C}$?
- Describe the image of the set $\{z=x+iy:x>0,y>0\}$ under the mapping $w=\frac{z-i}{z+i}$
- How does one define the complex distribution $1/z$?

As an improper integral, $\oint_{|z|=1} \frac{1}{z-1}\; dz$ does not converge. You could define a principal value which is the limit as $\epsilon \to 0+$ of the integral over arcs omitting a length $\epsilon$ on each side of the singularity. But you have to be careful doing such an integral using residues, because the arc you use to close up the contour will make a significant contribution.

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- Differentiability of $f(x) = \exp(-1/x^2), f(0) = 0$
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- Dot product for 3 vectors
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- Norm of a Kernel Operator
- Show that $2 < e^{1/(n+1)} + e^{-1/n}$