MATH3401

Lecture 20

Contour integrals

Further examples

Example 4: Evaluate   $$I = \ds \int_C \frac{dz}{z}$$, where $$C= 2e^{i\theta}$$, $$0\leq \theta \leq 2 \pi$$.

Can't repeat argument from Example 3 directly.

 $$I = I_1 + I_2$$ where $$I_j = \ds \int_{C_j}\frac{dz}{z}$$

Contour integrals

Further examples

Example 4 (cont): On $D = \C \setminus \{\text{negative real axis}\cup \{0\}\},$ $$\Log z$$ is a primitive for $$\dfrac{1}{z}$$. $$C_1\subset D$$, so Theorem in Lecture 19 $$\Rightarrow$$

$$I_1 = \Log(2i)-\Log(-2i)$$ $$= \ln|2i| + i\dfrac{\pi}{2}-\left(\ln|-2i| + i\left(-\dfrac{\pi}{2}\right)\right)$$

$$\quad = \pi i$$.

Remark: Agrees with calculation from Lecture 19, Example 1.

Contour integrals

Further examples

Example 4 (cont): For $$I_2$$, on $D' = \C \setminus \{\text{positive real axis}\cup \{0\}\},$ $$1/z$$ has a primitive, e. g. $$\mathcal \Log z = \ln|z| + i \,{\Large\mathcal a}\mathcal rg(z)$$, $$0 \leq {\Large\mathcal a}\mathcal rg(z) < 2 \pi$$.

Note $$C_2 \subset D'$$. So Theorem in Lecture 19 $$\Rightarrow$$

$$I_2 = \mathcal \Log(-2i)-\mathcal \Log(2i)$$ $$= \ln 2 + i\dfrac{3\pi}{2}-\left(\ln 2 + i\left(\dfrac{\pi}{2}\right)\right)$$

$$\quad = \pi i$$.

Thus $$I= I_1 + I_2 = 2 \pi i$$.

Contour integrals

Further examples

Example 4 (cont): So $\int_C z^n dz = \left\{ \begin{array}{lr} 0 & n\in \Z\setminus \{-1\}\\ 2 \pi i & n=-1 \end{array} \right.$

for any circle $$C$$ centred at the origin, positively oriented.

Cauchy-Goursat

Let $$C$$ be a simple closed curve in $$\C$$. If $$f$$ is analytic on $$C$$ and its interior, then $\int_C f\left(z\right)dz = 0.$

Cauchy-Goursat

Remark: $$\ds \int_C f\left(z\right)dz = 0$$ $$\nRightarrow$$ $$f$$ is analytic in and on $$C$$.

e. g. $$\ds \int_C z^n dz = 0$$, $$n= -2, -3, -4, \ldots$$

Proof of C-G: Do it for

Cauchy-Goursat

Key step in proof: $$M$$-$$\ell$$ estimate

Suppose $$f$$ is continuous on a contour $$C$$ given by $$z = z(t)$$, $$a\leq t \leq b$$. Then exists $$M$$ such that $|f\left(z\right)| \leq M \quad \forall z\in\C,$

via extreme value theorem in $$\R$$.

Cauchy-Goursat

So $$\ds \int_C f \left(z\right) dz =\left| \ds \int_a^b f \left(z \left(t\right)\right) z' \left(t\right) dt\right|$$

$$\qquad \qquad \quad \leq \ds \int_a^b \left| f \left(z \left(t\right)\right) \right| \cdot \left| z' \left(t\right)\right|dt$$

$$\qquad \qquad \quad \leq M \ds \int_a^b \left| z' \left(t\right)\right|dt$$ $$= M \ell$$.

where $$\ell = \ell\left(C\right)=\text{length}$$ of $$C$$.

Cauchy-Goursat

Recall: a domain $$D$$ is simply connected if for every simple closed contour $$C$$ in $$D$$, there holds $\text{Int}\, C \subset D.$

i. e. "no holes". In other words, all simple closed contours are null homotopic.

If $$D$$ is not simply connected, it is multiply connected.

Cauchy-Goursat extension

Multiply connected domains

If $$f$$ is analytic on simple closed contours $$C$$, $$C_1, C_2, \ldots, C_n\subset \text{Int}\, C$$, and the interior of the domain bounded by $$C_1, C_2, \ldots, C_n$$, with $$C$$ positively oriented and $$C_k$$ negatively oriented, then $\int_C f + \sum_{j=1}^{n} \int_{C_j} f = 0.$