# Thread: [SOLVED] Line integral, Cauchy's integral formula

1. ## [SOLVED] Line integral, Cauchy's integral formula

Let $\displaystyle C$ be the frontier of the square whose sides are the lines $\displaystyle x=\pm 2$ and $\displaystyle y= \pm 2$. (orientation is positive)
Calculate the following integral using Cauchy's integral formula for a function or its derivative.

$\displaystyle \int _C \frac{e^{-z}dz}{z-\frac{i\pi}{2}}$.

My attempt : I find confusing the "using Cauchy's integral formula for a function or its derivative.", I don't really know what they mean by that.

So Cauchy's formula is $\displaystyle f(z)=\frac{1}{2\pi i} \int _C \frac{f(\xi)}{\xi -z}d \xi$.

Looking at the integrand of the integral, I see that it is not defined for $\displaystyle z=i \frac{\pi}{2}$. Does that mean that $\displaystyle f$ is not analytic in this point?
I believe the integral should equals $\displaystyle 2 \pi i$, I don't really know why.

I need help, I find all this new theory somewhat hard to grasp due to the amount of it.

2. Cauchy's Integral Formula states that if $\displaystyle f(z)$ is analytic in $\displaystyle C$ and the point $\displaystyle z$ is not on C, then:

$\displaystyle f(z)=\frac{1}{2\pi i} \int _C \frac{f(\xi)}{\xi -z}d \xi$

Ok, $\displaystyle f(z)=e^{-z}$ and that's analytic in C. Don't confuse the ENTIRE integrand with the expression for $\displaystyle f(z)$ when using this formula.

So plugging in $\displaystyle z=\pi i/2$, then the integral is just:

$\displaystyle 2\pi i f(\pi i/2)$ right?

Try and get good at calculating these in Mathematica so that you can check your results. For example (where I used an equivalent circle of radius 2 via deformation of path theorem):

Code:
In[111]:=
NIntegrate[(Exp[-z]/(z - Pi*(I/2)))*2*I*
Exp[I*t] /. z -> 2*Exp[I*t],
{t, 0, 2*Pi}]
N[2*Pi*I*Exp[(-Pi)*(I/2)]]

Out[111]=
6.283185307179801 -
2.708944180085382*^-14*I

Out[112]=
6.283185307179586

3. Thanks for the reply, however I'm extremely confused.
Originally Posted by shawsend
Cauchy's Integral Formula states that if $\displaystyle f(z)$ is analytic in $\displaystyle C$ and the point $\displaystyle z$ is not on C, then:

$\displaystyle f(z)=\frac{1}{2\pi i} \int _C \frac{f(\xi)}{\xi -z}d \xi$
but $\displaystyle z$ has to be in the interior of $\displaystyle C$, right?
If so, then in the exercise I can consider $\displaystyle z$ as $\displaystyle i \frac{\pi}{2}$ and I notice that indeed $\displaystyle z$ is inside the square $\displaystyle C$.
Also z is a pole I think. I don't know if it's important.

Ok, $\displaystyle f(z)=e^{-z}$ and that's analytic in C. Don't confuse the ENTIRE integrand with the expression for $\displaystyle f(z)$ when using this formula.

So plugging in $\displaystyle z=\pi i/2$, then the integral is just:

$\displaystyle 2\pi i f(\pi i/2)$ right?
Here I don't understand why you worked with the numerator of the integrand only. I know that it is analytic in $\displaystyle C$. However I think the denominator is not analytic at $\displaystyle z$, inside $\displaystyle C$, the square.

Try and get good at calculating these in Mathematica so that you can check your results. For example (where I used an equivalent circle of radius 2 via deformation of path theorem):

Code:
In[111]:=
NIntegrate[(Exp[-z]/(z - Pi*(I/2)))*2*I*
Exp[I*t] /. z -> 2*Exp[I*t],
{t, 0, 2*Pi}]
N[2*Pi*I*Exp[(-Pi)*(I/2)]]

Out[111]=
6.283185307179801 -
2.708944180085382*^-14*I

Out[112]=
6.283185307179586
I never heard of such a theorem... looks interesting though.
Before computing I'd like to understand what I'm doing. I know much less than you think.

4. Oh... I think I just understood what you meant :
Thanks to CFI we have $\displaystyle 2 \pi i f(\frac{i \pi}{2})= \int _C \frac{e^{-z}dz}{z- \frac{i \pi}{2}}$.
But $\displaystyle f(\frac{i\pi}{2})=e^{-\frac{i \pi}{2}}\Rightarrow \int _C \frac{e^{-z}dz}{z- \frac{i \pi}{2}}=2\pi i \cdot (-i)= 2\pi$.