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A geometric description would be that it rotates z by an angle of theta in the counter clockwise direction. Correct?

Writing function in polar can I say -

$f(z) = e^{i\theta}re^{i\theta}$

or should that be -

$f(z) = e^{i\theta}re^{i\gamma}$

I know the $\theta$ is used in the definition of a complex number in polar form , $re^{i\theta}$, but it looks like in this case it is 'already taken' by the first number ($e^{i\theta}$)...or maybe the question means for me to use in the second number too? Or do I even need to bother writing out this stuff, will the once line answer above suffice?

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up vote 1 down vote accepted

The one line answer should be good enough for most, except perhaps in classes where great emphasis is placed on minutely detailed explanations. (Such requirements have their place, but they should not permeate an educational program.) When it is time to elaborate, as you have noticed the conventional $\theta$ in the polar form is already taken, so you must pick another letter, perhaps saying like this: Writing $z$ in polar form as $z=re^{i\gamma}$, we find $f(z)=re^{i(\theta+\gamma)}$, which is $z$ rotated an angle $\theta$.

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If you're asked to give a geometric description of the function $z \mapsto e^{i\theta}z$ then I don't think it's acceptable to just write that it sends $re^{i\gamma} \mapsto re^{i(\gamma + \theta)}$ $-$ that's a bit like rewriting the question. It sounds more like this question was intended to test understanding and intuition, probably in relation to the Argand diagram, and that the answer they're looking for is something along the lines of 'rotation by an angle $\theta$ about the origin'. – Clive Newstead Apr 7 '12 at 11:48
@CliveNewstead: That is what the final few words of my answer are for. Now I admit that my writing tends to be terse, but I see little point in going on and on about what it means to rotate $z$ around the origin. Unless the class is on didactics, in which case you could probably write a minor thesis on the various ways in which this notion could be misunderstood. – Harald Hanche-Olsen Apr 7 '12 at 12:03

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