← 2016 Paper 2

UPSC 2016 Maths Optional Paper 2 Q1d — Step-by-Step Solution

10 marks · Section A

Harmonic functions and harmonic conjugate · Complex Analysis · asked 7× in 13 yrs · Read the full method →

Question

Is $v(x,y)=x^3-3xy^2+2y$ a harmonic function? Prove your claim. If yes, find its conjugate harmonic function $u(x,y)$ and hence obtain the analytic function whose real and imaginary parts are $u$ and $v$ respectively.

Technique

Laplacian test for harmonicity; recover the conjugate by integrating one CR equation and fixing the integration “constant” $\varphi(y)$ via the other; identify $f(z)$ by spotting $z^3$ (and $iz^3$) and $z$ in $u\pm iv$.

Solution

We are given $v$ as the imaginary part. We verify $v$ is harmonic (so a conjugate exists on the simply connected plane), recover $u$ from the Cauchy–Riemann equations, and assemble $f=u+iv$ as a function of $z$.

Step 1 — Harmonicity of $v$

Compute the second partials of $v=x^3-3xy^2+2y$:

Then

So $v$ is harmonic on all of $\mathbb R^2$.

Step 2 — Set up Cauchy–Riemann for the conjugate $u$

For $f=u+iv$ to be analytic, $u$ and $v$ satisfy the Cauchy–Riemann equations

Hence

Step 3 — Integrate to find $u$

Integrate $u_x=-6xy+2$ with respect to $x$:

where $\varphi$ is a function of $y$ alone. Differentiate in $y$ and match $u_y$:

Therefore the conjugate harmonic function is

(One checks $u_{xx}+u_{yy}=-6y+6y=0$: $u$ is harmonic, as it must be.)

Step 4 — Assemble the analytic function $f=u+iv$

Recognize the cubic and linear pieces via $z=x+iy$:

Adding, $iz^3+2z=(-3x^2y+y^3+2x)+i(x^3-3xy^2+2y)=u+iv$. Hence

Answer