Lecture Notes

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====== Contour Integral ======
Reading: Boas, Ch14, section 1-5


So, you have learned what holomorphic function looks like, and you know there are functions which are 'bad' only at a few points. What do you want to do with these functions? 



===== primitive of a holomorphic function on $\C$ =====
Just like calculus, you can do differentiation, and you can do integration. Differentiation is easy, let's talk about integration. 

Recall what we do in real analysis case: given $f(x)$ on $\R$, we can find one primitive $F(x)$ by considering
$$ F(x) = C + \int_{x_0}^x f(u) du $$
where we set the initial condition that $F(x_0) = C$, and $F'(x)= f(x)$. 

Can we do the same here? Say $f(z)$ is a holomorphic function, we can define
$$ F(z) = C + \int_{z_0}^z f(u) du $$
Now, we immediately run into trouble: how do we go from $z_0$ to $z$? Does the integration depends on how we choose the path from $z_0$ to $z$? Thanks to the fact that $f$ is holomorphic, the integration is independent of the choice of path. 

===== primitive of $1/z$ =====
OK, $f(z) = 1/z$ is not a holomorphic function on the entire $\C$. We can say, it is a holomorphic function on the 'punctured complex plane' $\C^* = \C \RM \{0\}$, or it is a meromorphic function on $\C$ with a pole of order $1$ at $z=0$. Either way, we can ask, can we find the primitive of $1/z$ on $\C^*$? Namely, is there a hol'c function $F(z)$ such that $F'(z) = 1/z$? 

You probably know that, for $x>0$, if you integrate $1/x$, you get $\log (x) + C$. (why is that? )

The same holds for complex analytic function. almost. We can say the primitive of $1/z$ is $\log(z) + C$, but $\log(z)$ is a multivalued function on $\C^*$. 

===== Contour integral for meromorphic function =====
Let $f: \Omega \to \C$ be a meromorphic function. 
Let $\gamma$ be a closed contour in $\Omega$ ("contour" just means a smooth path) that avoids the pole of $f$. Then, 
$$ \int_\gamma f(z) dz = (2 \pi i) \sum_{z_0 \text{poles of } f} Res_{z_0} f. $$
The $Res_{z_0} f$ is the residue of $f$, which is the Laurent expansion of $f$ at $z$, the coefficient in front of $1/(z-z_0)$.

Example: $f(z) = 1/ [(z-1)(z-2)(z-3)]$, $\gamma$ is a contour  around the two poles $1$ and $2$.

===== Exercise =====
1. For $t \in [0, 2\pi]$, let $z(t) = e^{it}$. Compute $\int_{0}^{2\pi} 1 / (z(t)) d z(t). $

2. For $t \in [0, 2\pi]$, let $z(t) = e^{i2t}$. Compute $\int_{0}^{2\pi} 1 / (z(t)) d z(t). $

3. For $t \in [0, 2\pi]$, let $z(t) = e^{-it}$. Compute $\int_{0}^{2\pi} 1 / (z(t)) d z(t). $
Lecture Notes

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