Complex Functions | Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE) PDF Download
Q. 1 Given 
the value of the definite integral
[2015 : 1 Mark, Set-II]
(a) 1
(b) -1
(c) i
(d) - i
Ans: c
Q. 2 Consider the following complex function
Which of the following is one of the residues of the above function ?
[2015 : 2 Marks, Set- 1]
(a) -1
(b) 9/16
(c) 2
(d) 9
Ans: a
f(z) has poles at z = 1, - 2 Residue of f(z ) at ( z = 1)
Residue of f(z) at ( z = -2 )
Q. 3 
can be expressed as
(a) - 0.5 - 0.5i
(b) - 0.5 + 0.5i
(c) 0.5 - 0.5i
(d) 0.5 + 0.5i [2014 : 1 Mark, Set-II]
Ans: (b)
Q. 4 For an analytic function, f{x; + iy) = u(x, y) + iv[x, y), u is given by u = 3x2- 3y2 . The expression for v, considering K to be a constant is
(a) 3 y2 - 3x2 + K
(b) 6x - 6 y + K
(c) 6y - 6x + K
(d) 6xy + K [2011 : 2 Marks]
Ans (d)
f= u + iv
u = 3x2 - 3y2
For f to be analysis, we have Cauchy-Riemann conditions,
From (i) we have,
Now applying equation (ii) we get,
Substitute in equation (iii),
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FAQs on Complex Functions - Topic wise GATE Past Year Papers for Civil Engineering - Civil Engineering (CE)
| 1. What are complex functions? |  |
Ans. Complex functions are functions that take complex numbers as inputs and produce complex numbers as outputs. They can be expressed as f(z) = u(x, y) + iv(x, y), where z = x + iy is a complex number, u(x, y) is the real part of the function, and v(x, y) is the imaginary part of the function.
| 2. How are complex functions different from real functions? | |
Ans. Complex functions differ from real functions as they operate on complex numbers instead of real numbers. While real functions map real numbers to real numbers, complex functions map complex numbers to complex numbers. Complex functions can exhibit more complex behavior, such as rotational or spiraling patterns, which are not possible with real functions.
| 3. What is the concept of analyticity in complex functions? | |
Ans. Analyticity is a fundamental property of complex functions. A function is said to be analytic in a region if it has a derivative at every point within that region. Analytic functions possess many important properties, such as having a power series representation, being infinitely differentiable, and satisfying the Cauchy-Riemann equations.
| 4. What is the significance of Cauchy-Riemann equations in complex functions? | |
Ans. The Cauchy-Riemann equations play a crucial role in the study of complex functions. These equations provide necessary and sufficient conditions for a complex function to be analytic. They establish a relationship between the partial derivatives of the real and imaginary parts of a complex function, enabling us to determine if a given function is analytic or not.
| 5. How are complex functions used in real-world applications? | |
Ans. Complex functions find applications in various scientific and engineering fields. They are used in electrical engineering to analyze the behavior of alternating current circuits, in fluid dynamics to model the flow of fluids, in signal processing for analyzing complex signals, and in quantum mechanics to describe wavefunctions. Complex functions are also extensively used in complex analysis, a branch of mathematics that studies functions of complex variables.
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