a) Evaluate using Cauchy's integral formula $$ \int_{c} \frac{5^{2}}{\left(7-z^{2}\right)(x+20} d z \quad $$ where $$ C:|z|=4 $$ b) Prove that the function below satisfy Cauchy integral theorem. $$ f(z)=2 z^{2}+z+5 \text { where } \mathrm{C} \text { is a circle }|x|=5 \text {. } $$ c) Evaluate $$ \int_{(0,6)}^{(3,7)}\left(-5 y+2 x^{2}\right) d x+(6 x-2 y) d x $$ along the parabola $$ x=3 t, y=t^{2}+6 $$ d) Evaluate $$ \$ \frac{e^{4 z}}{(z-1)^{2}} d z $$ along $$ c:|z|=3 $$ by utilizing Cauchy integral formula. e) Show that the function $$ U(x, y)=x^{3}-3 x y^{2}+3 x^{2}-3 y^{2}+1 $$ is harmonic. Hence or otherwise find $$ v $$ such that $$ f=u+i v $$

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Spencer Delgado · Accepted Answer

a) Evaluate the integral using Cauchy's integral formula and find the residues at the singularities within the contour. b) Prove that the function $$ f(z) = 2z^2 + z + 5 $$ is analytic in the region enclosed by the contour $$ |z| = 5 $$ to satisfy the Cauchy integral theorem. c) Evaluate the line integral along the parabola $$ x = 3t, y = t^2 + 6 $$ by substituting into the integral and simplifying. d) Use the Cauchy integral formula for derivatives to evaluate the integral along $$ |z| = 3 $$. e) Show that the function $$ U(x, y) $$ is not harmonic and find $$ v $$ such that $$ f = u + iv $$ is analytic using the Cauchy-Riemann equations.

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