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UPSC 2020 Maths Optional Paper 2 Q1d — Step-by-Step Solution

10 marks · Section A

Cauchy's theorem (Cauchy-Goursat) · Complex Analysis · asked 2× in 14 yrs · Read the full method →

Question

Evaluate the integral $\int_C (z^2+3z)\,dz$ counterclockwise from $(2,0)$ to $(0,2)$ along the curve $C$, where $C$ is the circle $|z|=2$.

Technique

Fundamental theorem for contour integrals: an entire integrand has a global antiderivative, so $\int_C f=F(\text{end})-F(\text{start})$.

Solution

Step 1 — The integrand is entire, so the integral is path-independent

$f(z)=z^2+3z$ is a polynomial, hence entire (analytic on all of $\mathbb{C}$). For an analytic function with an antiderivative, the contour integral depends only on the endpoints:

The curve $C$ is the quarter-arc of $|z|=2$ traversed counterclockwise from $(2,0)$ to $(0,2)$ — that is, from $z_{\text{start}}=2$ to $z_{\text{end}}=2i$. (Counterclockwise on $|z|=2$ takes $2\to 2i$ through the first quadrant; the specific arc does not matter since $f$ is entire, but it is consistent here.)

Step 2 — Evaluate the antiderivative at the endpoints

Step 3 — Subtract

Answer