Fisicaexe - Solved Problems in Physics

Cauchy's Integral Formula

Find the integrals using the Cauchy's Integral Formula with the paths traversed counterclockwise

a) \( \displaystyle \oint_{|z|=2}\frac{z}{z-1}\;dz \)

Solution

b) \( \displaystyle \oint_{{|z|=2}}\frac{\operatorname{e}^{\pi\mathrm{i}z}}{z-1}\;dz \)

Solution

c) \( \displaystyle \oint_{{|z|=4}}\frac{z^{2}+1}{z+2}\;dz \)

Solution

d) \( \displaystyle \oint_{{|z|=1/2}}\frac{z^{2}+2z+3}{3z-1}\;dz \)

Solution

e) \( \displaystyle \oint_{{|z-\mathrm{i}|=2}}\frac{\operatorname{sen}z}{z-\frac{\pi}{4}}\;dz \)

Solution

f) \( \displaystyle \oint_{{|z|=1}}\frac{z^{2}}{z-2\mathrm{i}}\;dz \)

Solution

g) \( \displaystyle \oint_{|z|=4}\frac{z^{2}}{z-2\mathrm{i}}\;dz \)

Solution

h) \( \displaystyle \oint_{{|z|=2}}\frac{z}{\left(z^{2}-9\right)(z+\mathrm{i})}\;dz \)

Solution

i) \( \displaystyle \oint_{{|z-\mathrm{i}|=1}}\frac{\sin \;\frac{\mathrm{i}z\pi}{2}}{z^{2}+1}\;dz \)

Solution

Calculate the following integrals on the contours shown in the figures

a) \( \displaystyle \oint_{{C}}\frac{z}{(z+1)(z-2)}\;dz \)

Solution 1

Solution 2

b) \( \displaystyle \oint_{{C}}\frac{\operatorname{e}^{z}}{z(z-3)}\;dz \)

Solution

c) \( \displaystyle \oint_{{C}}\frac{\cos z}{z}\;dz \)

Solution

d) \( \displaystyle \oint_{{C}}\frac{z^{2}}{z-1-\mathrm{i}}\;dz \)

Solution

Find the integrals using the Cauchy's Integral Formula

a) \( \displaystyle \oint_{|z+\mathrm{i}|=1}\frac{\operatorname{sen}z}{(z+\mathrm{i})^{3}}\;dz \)

Solution

b) \( \displaystyle \oint_{|z-1|=1}\frac{dz}{(z-1)^{3}(z+1)^{3}} \)

Solution

c) \( \displaystyle \oint_{|z+1|=1}\frac{dz}{(z-1)^{3}(z+1)^{3}} \)

Solution

d) \( \displaystyle \oint_{|z|=3}\frac{\cos (z^{2}+3z-1)}{(2z+3)^{2}}\;dz \)

Solution

e) \( \displaystyle \oint_{|z|=1}\frac{z^{2}+z+\mathrm{i}}{(4z-\mathrm{i})^{3}}\;dz \)

Solution

f) \( \displaystyle \oint_{|z-2|=3}\frac{\operatorname{e}^{z}}{z^{3}(z-1)}\;dz \)

Solution