GATE EC 2014 Set 2 — Question 37
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Engineering Mathematics → Complex Analysis → Cauchy-Riemann Equations
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Question
The real part of an analytic function where is given by . The imaginary part of is
Correct answer
(B) e^(-y)sin(x)
Solution
Let . Given .
For to be analytic, it must satisfy the Cauchy-Riemann equations:
1)
2) From (1):
Integrating with respect to : From (2):
So, Differentiating our expression for with respect to :
Comparing the two expressions for , we get , which means is a constant.Thus, . Matching with the options, the imaginary part is .
For to be analytic, it must satisfy the Cauchy-Riemann equations:
1)
2) From (1):
Integrating with respect to : From (2):
So, Differentiating our expression for with respect to :
Comparing the two expressions for , we get , which means is a constant.Thus, . Matching with the options, the imaginary part is .
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