Laplace equation is called a harmonic function. Let the functions u = u(x, y) and v = v(x, y) be of class C² and satisfy the Cauchy–Riemann equations ди dv and dv du ду Show that u and v are both harmonic.

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Chapter4: Eigenvalues And Eigenvectors
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(3) The Partial Differential Equation
f, f
= 0
is called the Laplace equation. Any function f = (x, y) of class C² that satisfies the
Laplace equation is called a harmonic function. Let the functions u = u(x, y)
and v = v(r, y) be of class C2 and satisfy the Cauchy-Riemann equations
du
dv
dv
du
and
dy
Show that u and v are both harmonic.
Transcribed Image Text:(3) The Partial Differential Equation f, f = 0 is called the Laplace equation. Any function f = (x, y) of class C² that satisfies the Laplace equation is called a harmonic function. Let the functions u = u(x, y) and v = v(r, y) be of class C2 and satisfy the Cauchy-Riemann equations du dv dv du and dy Show that u and v are both harmonic.
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