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Question:
If \(f(z)\) is an analytic function of \(z\), \(\left( \frac{\partial^2 }{\partial x^2} + \frac{\partial^2 }{\partial y^2}\right) \left| f(z)\right|^2\) is equal to
| (A) \(2 \big| f^ \prime(z)\big|^2\) | (B) \(3 \big|f^\prime(z)\big|^2\) |
| (C) \(4 \big| f^\prime(z)\big|^2\) |
(D) \(8 \big| f^\prime(z)\big|^2\) |
Hint
Select a simple analytic function and test which option is correct.
Answer
Take \(f(z)=z\) and then it is easy to check that Option (C) is the correct option.
Full Solution
With choice \(f(z)=z\), we get \( \big|f(z)\big|^2 = (x^2+y^2)\).
Therefore
\(\left( \frac{\partial^2 }{\partial x^2} + \frac{\partial^2 }{\partial y^2}\right) \left| f(z)\right|^2 = 4\)
Also we have \(f^\prime(z)=1\).
Hence Option (D) is the correct option.
Acknowledgment
Communicated by Sunday-Physics-Participant
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Comments
From a participant of Sunday
From a participant of Sunday Physics
Sir i got the solution as 4 ,then i have confusion why |f'(z)|^2 ?
I guess you are asking why is
I guess you are asking why is there |f'(z)|^2 in the answer?
Then the answer is as follows. You have taken f(z) =z as test case.
You computed the expression in the main stem and you get the answer 4.
Next, you must also compute the expressions in different options for the same function f(z) =z.
This give f'(z)=1. The four options now ( for this test case) become
(A) 2 (B) 3 (C) 4 (D) 8.
So option (C) is the correct option.