MATH 406: Additional Problems for PS 9




  1. Evaluate the following integrals around the curve $ \gamma$ defined by $ \vert z\vert = 7$ :

       a) $\displaystyle \, \int_\gamma \frac{e^z + \sin z}{(z-i)^3} \, dz$   b) $\displaystyle \, \int_\gamma \frac{\cosh z}{z^2} \, dz$   c) $\displaystyle \, \int_\gamma \frac{1}{z^2 - 4z +1} \, dz
$

  2. Find

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




  3. Evaluate:

    $\displaystyle \int_{0}^{\pi} \frac{\cos \theta}{17 - 8 \cos \theta}\, d\theta
$




  4. Find the real and imaginary parts of :

       a) $\displaystyle \, F_3 = 39i + \bar{z}+ \bar{z}^2 z$   b) $\displaystyle \, F_2 = z + \bar{z}\cos z \,
$

  5. Show that the integral of a 2-analytic function

    $\displaystyle 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

    $\displaystyle f_1 = e^{\sin(z)}$   and$\displaystyle \qquad f_2 = e^z$







Andrew L. Belmonte 2005-03-22