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10. Show that where A is the Laplacian Əz dz = 88 az Əz Δ,

10. Show that where A is the Laplacian Əz dz = 88 az Əz Δ,

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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.
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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