GATE EE 2024 Set 1 — Question 30
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Engineering Mathematics → Complex Variables → Analytic Functions
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Question
Which of the following complex functions is/are analytic on the complex plane?
Correct answer
(D) f(z) = z² - z
Solution
A complex function is analytic if it satisfies the Cauchy-Riemann (CR) equations:
and .
Let .(A) .
Here, and .
, . (First CR equation satisfied)
, .
Since , the second CR equation is not satisfied. Thus, is not analytic.(B) .
Here, and .
, . (First CR equation satisfied)
, .
Since , the second CR equation is not satisfied. Thus, is not analytic.(C) .
Here, and .
for .
.
For the first CR equation to hold, , which implies . This is not true for all . Thus, is not analytic.(D) .
This is a polynomial in . Polynomials are entire functions, meaning they are analytic everywhere in the complex plane.
Let's verify using CR equations:
.
So, and .
.
.
The first CR equation is satisfied.
.
.
The second CR equation is satisfied ().
Since both CR equations are satisfied, is analytic on the complex plane.The final answer is .
and .
Let .(A) .
Here, and .
, . (First CR equation satisfied)
, .
Since , the second CR equation is not satisfied. Thus, is not analytic.(B) .
Here, and .
, . (First CR equation satisfied)
, .
Since , the second CR equation is not satisfied. Thus, is not analytic.(C) .
Here, and .
for .
.
For the first CR equation to hold, , which implies . This is not true for all . Thus, is not analytic.(D) .
This is a polynomial in . Polynomials are entire functions, meaning they are analytic everywhere in the complex plane.
Let's verify using CR equations:
.
So, and .
.
.
The first CR equation is satisfied.
.
.
The second CR equation is satisfied ().
Since both CR equations are satisfied, is analytic on the complex plane.The final answer is .
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