Exercise · ZW-HARMONIC-FUNCTION-003 · ★☆☆ Basic
A harmonic conjugate of \(u=x^2-y^2\) is
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- C. \(2xy\)
Explanation
We need \(v_y=u_x=2x\) and \(v_x=-u_y=2y\); \(v=2xy\) satisfies both, and \(x^2-y^2+2ixy=z^2\).
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-004 A harmonic conjugate of u=e^xcos y is
- ZW-HARMONIC-FUNCTION-005 Which of these functions can be the real part of a holomorphic function on all of ℂ?
- 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…