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The function f(z) of complex variable z = x + iy, where i = √−1, is given as f(z) = (x3 – 3xy2) + i v(x,y). For this function to be analytic, v(x,y) should be
  • a)
    (3xy2 – y3) + constant
  • b)
    (3x2y2 – y3) + constant
  • c)
    (x3 – 3x2y) + constant
  • d)
    (3x2y – y3) + constant
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
The function f(z) of complex variable z = x + iy, where i = √&mi...
The function f(z) of complex variable z = x + iy, where i = √(-1), can be written as f(z) = u(x, y) + iv(x, y), where u(x, y) and v(x, y) are real-valued functions of x and y.

For example, if f(z) = z^2, then we can write it as f(z) = (x + iy)^2 = x^2 + 2ixy - y^2. In this case, u(x, y) = x^2 - y^2 and v(x, y) = 2xy.

In general, the real part u(x, y) represents the real-valued function associated with the real component of f(z), and the imaginary part v(x, y) represents the real-valued function associated with the imaginary component of f(z).
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The function f(z) of complex variable z = x + iy, where i = √&mi...
Concept:
f(z) = u + iv
u = real part
v = imaginary part
If f(z) is an analytic function

(This is an exact differential equation)
Calculation:
Given,
u = x3 – 3xy2
∂u/∂x = 3x2 − 3y2
∂u/∂y = −6xy

It is an exact differential equation the solution is obtained by treating y as constant in the first term and in the second term only that part is integrated which is not containing x.
Integrating the above equation
v = 3x2y − y+ constant
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The function f(z) of complex variable z = x + iy, where i = √−1, is given as f(z) = (x3 – 3xy2) + i v(x,y). For this function to be analytic, v(x,y) should bea)(3xy2 – y3) + constantb)(3x2y2 – y3) + constantc)(x3 – 3x2y) + constantd)(3x2y – y3) + constantCorrect answer is option 'D'. Can you explain this answer?
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The function f(z) of complex variable z = x + iy, where i = √−1, is given as f(z) = (x3 – 3xy2) + i v(x,y). For this function to be analytic, v(x,y) should bea)(3xy2 – y3) + constantb)(3x2y2 – y3) + constantc)(x3 – 3x2y) + constantd)(3x2y – y3) + constantCorrect answer is option 'D'. Can you explain this answer? for Civil Engineering (CE) 2024 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about The function f(z) of complex variable z = x + iy, where i = √−1, is given as f(z) = (x3 – 3xy2) + i v(x,y). For this function to be analytic, v(x,y) should bea)(3xy2 – y3) + constantb)(3x2y2 – y3) + constantc)(x3 – 3x2y) + constantd)(3x2y – y3) + constantCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The function f(z) of complex variable z = x + iy, where i = √−1, is given as f(z) = (x3 – 3xy2) + i v(x,y). For this function to be analytic, v(x,y) should bea)(3xy2 – y3) + constantb)(3x2y2 – y3) + constantc)(x3 – 3x2y) + constantd)(3x2y – y3) + constantCorrect answer is option 'D'. Can you explain this answer?.
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