MATH 406: Additional Problems for PS 9

1. Evaluate the following integrals around the curve $\gamma$ defined by $\vert z\vert = 7$ :
      a) $$\, \int_\gamma \frac{e^z + \sin z}{(z-i)^3} \, dz$$   b) $$\, \int_\gamma \frac{\cosh z}{z^2} \, dz$$   c) $$\, \int_\gamma \frac{1}{z^2 - 4z +1} \, dz$$

2. Find
      Res$$\left(\frac{e^{\pi z}}{(z-z_0)^3};z_0\right)$$

   
   
   
   

3. Evaluate:
   $$\int_{0}^{\pi} \frac{\cos \theta}{17 - 8 \cos \theta}\, d\theta$$

   
   
   
   

4. Find the real and imaginary parts of :
      a) $$\, F_3 = 39i + \bar{z}+ \bar{z}^2 z$$   b) $$\, F_2 = z + \bar{z}\cos z \,$$

5. Show that the integral of a 2-analytic function
   $$F_2 (z,\bar{z}) = f_1(z) + \bar{z}f_2(z)$$
   around a closed simple loop $\gamma$ is not identically zero (note that $f_1$ and $f_2$ are analytic). Evaluate the contour integral explicitly for the curve defined by $\vert z\vert = 2$ , with
   $$f_1 = e^{\sin(z)}$$   and$$\qquad f_2 = e^z$$

   
   
   
   

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