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I'm trying to teach myself complex analysis (I didn't have it in undergrad, and am doing my master's in France, where they all did have it in undergrad) with a book online, and one of the first exercises is causing me problems, as it isn't covered in the chapter. It's pretty trivial, so I probably should have learned it in some class a long time ago (maybe even in high school??), but I've never actually had to solve an equation like this. Here's the problem:

Find all solutions of $z^2 + 2z + (1-i) = 0$.

The answers given at the end of the book are in exponential form. I don't know many methods solving in this form, as I just learned a simple one from youtube.

Again, I'm sure it's terribly trivial, but if someone could show me how to solve this, that'd be great.

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    I studied at$a$well-ranked public university in the midwest. They had complex analysis, but for a bachelor's degree it was a choice, and I didn't take it. I also never had topology (except a tiny bit included in Real Analysis) and taught it to myself in January before my first semester of grad school. In retrospect, I should have taken both, but I didn't realize it would be so important.2012-07-11

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You can complete the square, and in case you knew what $e^z$ behaves like, here's a solution:

$z^2+2z+(1-i)=z^2+2z+1-i=(z+1)^2-i=0$ so that $(z+1)^2=i=e^{\frac{i\pi}2}$ hence $z+1=e^{\frac{i\pi}4}\,\,\vee\,\, e^{\frac{5i\pi}4}$ $z=1+e^{\frac{i\pi}4}\,\,\vee\,\,1+e^{\frac{5i\pi}4}$

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    Thank you, I had gotten z= -1 +/- (i)^1/2 from the quadratic formula, but wasn't sure how to put it into exponential form. (I see now).2012-07-11
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Hint $\ $ By the quadratic formula $\rm\: z = -1 \pm \sqrt{\it i\,}.\:$ By my Simple Denesting Rule, up to sign,

$ \sqrt{\it i\,}\, =\, \frac{{\it i} - 1}{\sqrt{-2}}\, =\, \alpha\,(1 + {\it i}),\ \ \ \alpha :=\frac{\sqrt{2}}2 =\, 0.7071\ldots$

Therefore $\rm\: z\, =\, -1\pm\sqrt{{\it i}\,}\,=-1\pm \alpha\,(1 + {\it i})\, =\, -1\pm \alpha\, \pm\, \alpha\,{\it i}$

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    @JKH Ah, I missed that. In any case, now you know how to do it a couple ways.2012-07-11