# MATH3401

Lecture 19

### Contour integral

Consider $$C$$ a contour $$z(t)$$, $$\,t\in [a,b]$$, $$\,z_1=z(a)$$, $$\,z_2=z(b)$$;
and $$f$$ is pwc on $$C$$.

Properties:

$$(*)$$ Linearity

• $$\alpha \in \C:$$ $$\ds \int_C \left(\alpha \, f \right) \left(z\right) dz = \alpha \int_C f \left(z\right) dz$$.
• $$\ds\int_C \left(\, f + g\right) \left(z\right) dz = \int_C f \left(z\right) dz + \int_C g \left(z\right) dz$$.
• $$C_1+ C_2$$ defines a contour when end point of $$C_1$$ is equal to start point of $$C_2$$.

### Contour integral

Properties (cont):

$$(*)$$ Given a contour $$C$$, we define $$-C$$ as follows: $w(t) = z(-t), \quad -b\leq t \leq -a.$

Then (check with change of parameter formula 📝): $\int_{-C} f\left(z\right) dz = - \int_{C} f\left(z\right) dz.$

Hence $$C_1 - C_2 = C_1 + \left(-C_2\right)$$ is defined when end point of $$C_1$$ is equal to start point of $$-C_2$$, which is equal to end point of $$C_2$$.

### Contour integral

Example 1: Evaluate   $$I=\ds \int_C \conj{z}\, dz$$   where $C: z=2e^{i\theta}, -\pi/2\leq \theta \leq \pi /2$

$$C$$ is pwc with respect to this parametrisation (indeed differentiable) and $$f$$ is continuous on $$C$$.

### Contour integral

Example (cont)

Note that $$z'(\theta) = 2 i e^{i\theta}$$. Then

$$I = \ds \int_{-\pi/2}^{\pi/2} f\left( z\left(\theta \right)\right) z'\left(\theta \right) d\theta$$

$$\;\;= \ds \int_{-\pi/2}^{\pi/2} \conj{\left( 2 e^{i\theta}\right)} \cdot 2i e^{i\theta} d\theta$$

$$\;\;= 4i \ds \int_{-\pi/2}^{\pi/2} e^{-i\theta} e^{i\theta} d\theta$$ $$=4\pi i$$.

### Contour integral

On $$C$$, $$z\conj{z} = 4$$. That is $$\conj{z} = \dfrac{4}{z}$$ Then $\int_C \frac{dz}{z}= \pi i.$

### Anti-differentiation

Let $$D$$ be a domain in $$\C$$ (i.e. an open, connected subset of $$\C$$).

An anti-derivative of $$f$$ on $$D$$ is $$F$$ such that $F'(z) = f(z)\text{ on } D.$

### Anti-differentiation

Theorem: The following are equivalent

1. $$f$$ has an anti-derivative on $$D$$;
2. For any $$z_1,z_2 \in D$$ and any contour $$C$$ from $$z_1$$ to $$z_2$$ in $$D$$, we have that $$\int_C f\left(z\right) dz$$ is independent of $$C$$;
3. For any closed contour $$C$$ in $$D$$ there holds $\int_C f\left(z\right)dz = 0.$

### Anti-differentiation

Proof: (i) $$\Rightarrow$$ (ii) follows from the fundamental theorem.

(ii) $$\Rightarrow$$ (iii): Let $$C$$ be a closed contour in $$D$$ with $$z(a) = z(b) = z_1$$. Fix $$\gamma \in (a,b)$$ such that $$z(\gamma)\neq z_1$$ and define contours in $$D$$ $C_1: \;\; w(t) = z(t), \quad a\leq t \leq \gamma\\ C_2: \;\; w(t) = z(t), \quad \gamma\leq t \leq b$

### Anti-differentiation

(ii) $$\Rightarrow$$ (iii) (cont): Then $$C_1+C_2 = C$$ and

$$\ds \int_C f$$ $$=\ds \int_{C_1+C_2} f$$ $$=\ds \int_{C_1} f + \int_{C_2} f$$

$$\qquad=\ds \int_{C_1} f - \int_{-C_2} f$$ $$=0$$, by (ii). $$\square$$

(iii) $$\Rightarrow$$ (ii) $$\Rightarrow$$ (i) see B/C.

### Anti-differentiation

Note in particular for $$C: z_1\ra z_2$$ in $$D$$ under (i) $$\Rightarrow$$ (iii),
there folds: $\int_Cf\left(z\right) dz = F(b)-F(a),$ where $$F$$ is ANY anti-derivative of $$f$$.

### Contour integrals

Further examples

Example 2: Evaluate   $$I=\ds \int_{0}^{1+i} z^2\, dz$$.

$$f(z)= z^2$$ is continuous on $$\C$$ and $$F(z) = \dfrac{z^3}{3}$$ is an anti-derivative on $$\C$$.

Theorem $$\Rightarrow I = F(1+i) -F(0)$$ $$=\frac{2}{3} (-1+i)$$.

### Contour integrals

Further examples

Example 3: Evaluate   $$I=\ds \int_C \frac{dz}{z^2}$$, where $$C= 2e^{i\theta}$$, $$0\leq \theta \leq 2 \pi$$.

$$f(z)= \dfrac{1}{z^2}$$ has an anti-derivative on $$\C_*$$, namely $$-\dfrac{1}{z}$$, and $$C$$ is a closed contour lying completely in $$\C_*$$,

Theorem $$\Rightarrow I = 0$$.

Indeed, same argument works to show $\int_C z^n \,dz = 0, \quad \forall n\in \Z\setminus \{-1\}.$