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Let $γ\colon[-1,1]\to\mathbb{C}$ , $γ(t)= z_0 + itc$ , $z_0$ fixed and c>0

Prove for x>0 $$\lim_{x\to0} \frac{1}{2πi} \int_γ \left(\frac{1}{z-w} - \frac{1} {z-w'}\right)dz = -1$$

Where $w=z_0 + x$ , $w'= z_0 - x$

I understand that you have to substitute in w and w' but I can't figure out what to do with $$\frac{1}{z-z_0 -x} - \frac{1}{z-z_0 +x}$$ What is the next step? Do I Use the $γ(t)$ function?

Any help on this question would be appreciated. Thanks

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Please refer to the section FAQ and use LaTeX to properly write mathematics in this site. I tried to edit your post but it confuses me, in particular> where you write "lim" do you meant of what going to what? And what is the function in the first integral, in particular that weird-looking " 1-/z-w' "...what is this? – DonAntonio Nov 20 '12 at 11:44
Hi, I've edited it now as you can see x goes to 0 for the limit. Hopefully now it is understandable, any help would be great. Thanks – CJC Nov 20 '12 at 12:23
Very nice, @Conor, yet I'm still wandering what did you actually mean by $\,z=z_0+itc\,$? Why did you write in that funny way the imaginary part? Perhaps you actually meant a circle of some radius (...$c$?) around the point $\,z_0\,$ in the complex plane? And what is $\,x\,$ after that in $\,w\,$? – DonAntonio Nov 20 '12 at 12:28
Z_0 s fixed in C(complex numbers) and c>0, xeR and x>0, I know that I have to substitute w and w' into the integral which I have, but I then think I have to use the γ(t) formula somehow – CJC Nov 20 '12 at 14:02
up vote 0 down vote accepted

Some hints:

Write $a>0$ instead of $x$. You may assume $z_0=0$. Then we have to look at the integral $$\int_{-ic}^{ic}\left({1\over z-a}-{1\over z+a}\right)\ dz\ ,$$ where the integration is along the segment $\sigma$ on the imaginary axis connecting $-ic$ and $ic$. Now use the fact that $${dz\over z-a}=d\log\bigl(|z-a|\bigr)+i\ d\arg(z-a)\ .$$ As $|ic\pm a|=|{-ic}\pm a|$ we only have to check how $\arg(z\pm a)$ changes along $\sigma$. This can be described in terms of the angles in the triangle with vertices $-ic$, $ic$, $a$, resp., $-ic$, $ic$, $-a$. (Draw a figure!)

Finally, analyze what happens when $a\to {0+}$ for fixed $c>0$.

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Thank you for your help. – CJC Dec 2 '12 at 17:03

Making the substitution $z=z_0+itc$ in the second integral gives

$$\int_\gamma\frac{1}{z-z_0+x}dz = \int_{-1}^1\frac{1}{itc+x}ic\,dt = \Bigl[\log(itc+x)\Bigr]_{-1}^1 = \log(x+ic)-\log(x-ic).$$

Using the principal value of $\log$,

$$\log(x \pm ic) = \frac12\log(x^2+c^2) \pm i\tan^{-1}(c/x)$$


$$\int_\gamma\frac{1}{z-z_0+x}dz = 2i\tan^{-1}(c/x) \to \pi i \text{ as }x \to 0^+.$$

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Thank you for the help. – CJC Dec 2 '12 at 17:04

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