Exercise · ZW-HARMONIC-FUNCTION-005 · ★☆☆ Basic
Which of these functions can be the real part of a holomorphic function on all of \(\mathbb{C}\)?
Choose one option.
Show answer
- B. \(x^3-3xy^2\)
Explanation
The real part of a holomorphic function is harmonic. \(x^3-3xy^2=\operatorname{Re}z^3\) is; the others have Laplacian \(2y\), \(4\) and \(-3e^x\cos 2y\).
Review the concept: Harmonic function
A real function \(u\) of two variables, with continuous second partial derivatives on an open set \(U\) of the plane, is harmonic if its Laplacian \(\Delta u\) is zero throughout \(U\).
\[\Delta u=\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0\]
More practice problems: harmonic function
- ZW-HARMONIC-FUNCTION-001 Which of these functions is harmonic on the whole plane?
- ZW-HARMONIC-FUNCTION-002 Compute Delta u for u=x^3+y^3.
- ZW-HARMONIC-FUNCTION-003 A harmonic conjugate of u=x^2-y^2 is
- ZW-HARMONIC-FUNCTION-004 A harmonic conjugate of u=e^xcos y is
- ZW-HARMONIC-FUNCTION-006 Find a so that u=x^2+ay^2 is harmonic.
- ZW-HARMONIC-FUNCTION-007 Find a>0 so that u=e^axsin y is harmonic.
- ZW-HARMONIC-FUNCTION-008 Find a harmonic conjugate of u=x^3-3xy^2.
- ZW-HARMONIC-FUNCTION-009 The function u=ln (x^2+y^2) is harmonic on ℂsetminus\0\. Does it have a harmonic conjugate on that…