MATH 406: Problem Set 11


due Friday, April 15, 2005


  1. (F 2.6.18) Compute the following integral, using the change of variables $ x=e^t$ to move the lower limit (and the singularity) to $ -\infty$ :

    $\displaystyle \int^{\infty}_{0}\,
\frac{(\ln x)^{2}}{x^2 + 1}\, dx
$

  2. Compute the following integral:

    $\displaystyle \int^{\infty}_{0}\,
\frac{x^{1/3}}{x^3 + 1}\, dx
$

  3. Fisher 3.3.5(d)

  4. Tell if each function is $ N$ -analytic, specifying $ N$ , then if the function is 2-analytic, check that the real and imaginary parts are biharmonic:

       a) $\displaystyle \, F = 7i z + \bar{z}+ \bar{z}^2 z^2$   b) $\displaystyle \, F = e^z + \vert z\vert^2$   c) $\displaystyle \, F = (z-12)(z+2i)(\bar{z}-1) \,
$

  5. Decide if the following functions are harmonic or not by checking the first Cauchy-Riemann equation $ u_x = v_y$ , then check if the functions is biharmonic by checking if the difference $ u_x - v_y$ is harmonic (as we showed in class):

       a) $\displaystyle \, \bar{z}z^2 + 16 \bar{z}^2$   b) $\displaystyle \, \bar{z}\cos z$   c) $\displaystyle \, z e^{\bar{z}}
$

  6. Based on the previous question, is a biharmonic function also harmonic, or not? Explain.




  7. Find all points at which the following mappings would not be conformal:

       a) $\displaystyle \, w = E z^2 + k_0$   b) $\displaystyle \, w = z\cos 3 z$   c) $\displaystyle \, w = z + \frac{1}{z}$   d) $\displaystyle \, w = \frac{z^2+3}{z^2+i} \,
$

  8. Fisher 3.4.8