10. Show that where A is the Laplacian Əz dz = 88 az Əz Δ,
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Transcribed Image Text:log z = log r + i
is holomorphic in the region r>0 and - < 0 <.
10. Show that
where z = re¹ with - < 0 <
where is the Laplacian
Ә
əz əz
A
= 4
əz əz
dx²
8²
dy²
11. Use Exercise 10 to prove that if f is bolomorphic in the open set , then the
real and imaginary parts of f are harmonic; that is, their Laplacian is zero.
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