GATE ME 2019 Set 1 — Question 37
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Engineering Mathematics → Complex Variables → Analytic Functions
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Question
A harmonic function is analytic if it satisfies the Laplace equation. If is a harmonic function, then its conjugate harmonic function is
Correct answer
(A) 4xy - 2x² + 2y² + constant
Solution
Given . For to be the conjugate harmonic function, the Cauchy-Riemann equations must be satisfied:
1)
2) From (1):
So, Now, differentiate with respect to :
From (2):
So,
Substituting back into the expression for :
This matches option (A).
1)
2) From (1):
So, Now, differentiate with respect to :
From (2):
So,
Substituting back into the expression for :
This matches option (A).
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