Determine which function is harmonic in $\mathbb R^2$:
$$\text{a) } y^2 \qquad \text{b) }x^2 + y^2\qquad \text{c) } e^x\qquad \text{d) }\operatorname{Im}((x + iy)^5)$$
I had this question come up in an exam, so I checked the first three functions, which are easier to differentiate, and by exclusion I determined that the last one was the only harmonic function.
However, to be really sure I wrote the Pascal triangle up to the sixth row, calculated the binomial power and differentiated the last one as well. It did indeed prove to be harmonic, but I feel like I wasted a lot of time.
Is there a faster way?
Hint: The real part and imaginary part of a holomorphic function are both harmonic.