complex analysis question

**shaheen7** · March 25th 2011, 04:05 PM

one of the review questions that will help me study for my upcoming exam is as follows.

Find all points z in C where the function h(z) = (z + 3)^2+i is differentiable. Find all points where this function is holomorphic. Assume that h is the principal value of the complex exponent.

Any help would be appreciated since I am quite confused.

**HallsofIvy** · March 25th 2011, 04:58 PM

Originally Posted by shaheen7

one of the review questions that will help me study for my upcoming exam is as follows.

Find all points z in C where the function h(z) = (z + 3)^2+i is differentiable. Find all points where this function is holomorphic. Assume that h is the principal value of the complex exponent.

Any help would be appreciated since I am quite confused.

The question is either trivial or doesn't make any sense. (z+ 3)^2+ i is a polynomial and all polynomials are differentiable for all z. On the other hand, "holomorphic" means "analytic for all z" so it makes no sense to ask for points on which a function "is holomorphic".

**shaheen7** · March 25th 2011, 05:56 PM

For the holomorphic part won't we need to use the Cauchy-Riemann equations? How would we go about finding du/dx, dv,dy and dv,dx, -dv,dy

**chisigma** · March 25th 2011, 10:27 PM

Originally Posted by shaheen7

one of the review questions that will help me study for my upcoming exam is as follows.

Find all points z in C where the function h(z) = (z + 3)^2+i is differentiable. Find all points where this function is holomorphic. Assume that h is the principal value of the complex exponent.

Any help would be appreciated since I am quite confused.

For semplicity sake we set $s=z+3$ and neglect the constant term i so that the function becomes $h(s)=s^{2}$ . Setting $s=\sigma + i\ \omega$ You find that is...

$h(s) = u(\sigma, \omega) + i\ v(\sigma, \omega) = \sigma^{2}- \omega^{2} + 2\ i\ \sigma\ \omega$

... and now it is easy to verify that the Cauchy Riemann equations are satisfied for any value of s...

Kind regards

$\chi$ $\sigma$

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