# Tagged Questions

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### Mean value theorem for harmonic

In Problems and Solution in Mathematics by Ta-Tsien, exercise 5123, the mean value theorem is used as: \text{log} |F(0)| = \frac{1}{2 \pi} \int_0^{2\pi} \text{log}|F(re^{i\theta})| ...
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### question related with cauch'ys inequality

We have a statement that if f(z) is analytic in a domain D and $C = \{z : |z-a| = R\}$ is contained in D. Then, $|f^n(a)|\leq \frac{n!M_R}{R^n}$ where $M_R = \max|f(z)|$ on C. What if the given D ...
I was doing problems on complex integration and got stuck at one question. The question is $$\int_{\gamma}{{\rm e}^{{\rm i}\pi z}\left(z + i\right)^{2}\cos\left(nz\right) ... 1answer 48 views ### Integrating Real Function in the Complex Plane Question: Evaluate the integral$$\int_{-\infty}^{\infty}\frac{\sin(x)}{x(x^2+a^2)}=Im\left ( \frac{e^{ix}}{x(x^2+a^2)} \right)$$... 4answers 146 views ### Evaluate \oint_C\frac{dz}{z-2} around the circle |z-2| = 4 I don't completely understand how to approach these questions. I suppose the notation \oint_C is something I'm not used to. So far, I have \oint_C\frac{dz}{z-2} = \log(z-2). From here, I suppose ... 1answer 105 views ### Evaluate \oint_C |z|^2 dz around the square with vertices at (0,0), (1,0), (1,1), (0,1) I don't think I quite understand how to go about this. My solution so far: \oint_C |z|^2 dz = \oint_C (x^2 + y^2)dz = \oint_C (x^2 + y^2) d(x+iy) = \oint_C x^2 + y^2 dx + i\oint_Cx^2+y^2dy. ... 2answers 2k views ### How to integrate complex exponential?? Consider$$\int^{\frac{1}{2}}_{-\frac{1}{2} } e^{i2\pi f} \,df = \int^{\frac{1}{2} }_{-\frac{1}{2} } \cos(2 \pi f)\, df$$Why do we only look at the real part? What about the imaginary part ... 2answers 104 views ### Computing with Cauchy Residue theorem how do I calculate$$\operatorname{Res}\left(\frac{1}{z^2 \cdot \sin(z))}, 0\right) What is the order of the pole? $3$?
Just as complex form of green's theorem $\int {f(z)}dz=i\int\int \frac{\partial f}{\partial x} + i\frac{\partial f}{\partial y}dxdy$ where $z=x+iy$ , do we have complex form of gauss divergence ...