← 2023 Paper 2

UPSC 2023 Maths Optional Paper 2 Q1d — Step-by-Step Solution

10 marks · Section A

Cauchy-Riemann equations (necessary and sufficient) · Complex Analysis · asked 5× in 13 yrs · Read the full method →

Question

State the sufficient conditions for a function $f(z)=f(x+iy)=u(x,y)+iv(x,y)$ to be analytic in its domain. Hence, show that $f(z)=\log z$ is analytic in its domain and find $\dfrac{df}{dz}$.

Technique

Verify (i) continuity of partials, (ii) Cauchy–Riemann; then $f'=u_x+iv_x$.

Solution

Part 1 — Sufficient conditions

Theorem. Let $\Omega\subseteq\mathbb{C}$ be open. The function $f=u+iv$ is analytic on $\Omega$ if, on $\Omega$:

(i) the partial derivatives $u_x,u_y,v_x,v_y$ exist and are continuous, and

(ii) the Cauchy–Riemann equations hold:

Under these hypotheses the complex derivative exists and equals

Part 2 — $\log z$ is analytic on its domain

Use the principal branch on $\Omega=\mathbb{C}\setminus(-\infty,0]$. Write $z=x+iy=re^{i\theta}$ with $r=|z|>0$ and $\theta\in(-\pi,\pi)$. Then

so

Step 1 — Partials of $u$ and $v$.

For $v$, on $\Omega$ the principal argument satisfies (locally, away from the negative real axis)

(Same expressions arise on the right and left half-planes after branch-consistent differentiation; the slit removes the discontinuity of $\arg$.)

Step 2 — Verify continuity.

All four partials are rational in $x,y$ with denominator $x^{2}+y^{2}$, which is non-zero on $\Omega$ (since $z\ne 0$). Hence the partials are continuous on $\Omega$.

Step 3 — Verify Cauchy–Riemann.

By the sufficient conditions, $f(z)=\log z$ is analytic on $\Omega=\mathbb{C}\setminus(-\infty,0]$.

Step 4 — Compute $df/dz$.

Answer