GATE ME 2020 Set 1 — Question 46
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Engineering Mathematics → Complex Variables → Analytic Functions
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Question
An analytic function of a complex variable () is defined as where is a real function. The value of the imaginary part of at is ________ (round off to 2 decimal places).
Correct answer
1.99 to 2.01
Solution
Let , where and .
Since is analytic, it must satisfy the Cauchy-Riemann equations:
1)
2)
From (1):Integrating with respect to :From (2):Differentiating our expression for with respect to :Equating the two expressions for :So, . For the standard analytic function , .
At , we have and .
The imaginary part is:The value is 2.00.
Since is analytic, it must satisfy the Cauchy-Riemann equations:
1)
2)
From (1):Integrating with respect to :From (2):Differentiating our expression for with respect to :Equating the two expressions for :So, . For the standard analytic function , .
At , we have and .
The imaginary part is:The value is 2.00.
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