GATE ME 2018 Set 1 — Question 14
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Engineering Mathematics → Complex Variables → Cauchy-Riemann Equations
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Question
F(z) is a function of the complex variable given by For what value of will satisfy the Cauchy-Riemann equations?
Correct answer
(B) 1
Solution
The Cauchy-Riemann equations for a complex function are:
Given .
Substitute , , .
From this, we identify the real part and the imaginary part :
Now, we find the partial derivatives:
Apply the first Cauchy-Riemann equation:
Apply the second Cauchy-Riemann equation:
This second equation is satisfied for any value of .For to satisfy the Cauchy-Riemann equations, must be 1.The final answer is
Given .
Substitute , , .
From this, we identify the real part and the imaginary part :
Now, we find the partial derivatives:
Apply the first Cauchy-Riemann equation:
Apply the second Cauchy-Riemann equation:
This second equation is satisfied for any value of .For to satisfy the Cauchy-Riemann equations, must be 1.The final answer is
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