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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?
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Most Upvoted Answer
The function f(z) of complex variable z = x + iy, where i = √&mi...
Understanding the Analytic Function
For a function f(z) = u(x,y) + iv(x,y) to be analytic, it must satisfy the Cauchy-Riemann equations. Here, u(x,y) = x^3 - 3xy^2, and we need to find v(x,y) that makes f(z) analytic.
Cauchy-Riemann Equations
The Cauchy-Riemann equations are given by:
- ∂u/∂x = ∂v/∂y
- ∂u/∂y = -∂v/∂x
Calculating partial derivatives of u:
- ∂u/∂x = 3x^2 - 3y^2
- ∂u/∂y = -6xy
Now, we can set up the equations:
From ∂u/∂x
- We have ∂v/∂y = 3x^2 - 3y^2.
From ∂u/∂y
- We have -∂v/∂x = -6xy, which implies ∂v/∂x = 6xy.
Finding v(x,y)
To find v(x,y), we integrate the equations:
1. Integrating ∂v/∂y = 3x^2 - 3y^2 gives:
- v(x,y) = (3x^2y - y^3) + C(x), where C(x) is an arbitrary function of x.
2. Next, we differentiate v with respect to x:
- ∂v/∂x = 6xy + C'(x).
Setting this equal to our earlier derived expression ∂v/∂x = 6xy, we find that C'(x) must be zero, meaning C(x) is a constant.
Conclusion
Thus, the function v(x,y) that satisfies the Cauchy-Riemann equations and makes f(z) analytic is:
v(x,y) = 3x^2y - y^3 + constant.
Therefore, the correct option is d) (3x^2y - y^3) + constant.
For a function f(z) = u(x,y) + iv(x,y) to be analytic, it must satisfy the Cauchy-Riemann equations. Here, u(x,y) = x^3 - 3xy^2, and we need to find v(x,y) that makes f(z) analytic.
Cauchy-Riemann Equations
The Cauchy-Riemann equations are given by:
- ∂u/∂x = ∂v/∂y
- ∂u/∂y = -∂v/∂x
Calculating partial derivatives of u:
- ∂u/∂x = 3x^2 - 3y^2
- ∂u/∂y = -6xy
Now, we can set up the equations:
From ∂u/∂x
- We have ∂v/∂y = 3x^2 - 3y^2.
From ∂u/∂y
- We have -∂v/∂x = -6xy, which implies ∂v/∂x = 6xy.
Finding v(x,y)
To find v(x,y), we integrate the equations:
1. Integrating ∂v/∂y = 3x^2 - 3y^2 gives:
- v(x,y) = (3x^2y - y^3) + C(x), where C(x) is an arbitrary function of x.
2. Next, we differentiate v with respect to x:
- ∂v/∂x = 6xy + C'(x).
Setting this equal to our earlier derived expression ∂v/∂x = 6xy, we find that C'(x) must be zero, meaning C(x) is a constant.
Conclusion
Thus, the function v(x,y) that satisfies the Cauchy-Riemann equations and makes f(z) analytic is:
v(x,y) = 3x^2y - y^3 + constant.
Therefore, the correct option is d) (3x^2y - y^3) + constant.
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Community Answer
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)
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
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 − y3 + constant
v = 3x2y − y3 + 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 Electrical Engineering (EE) 2024 is part of Electrical Engineering (EE) preparation. The Question and answers have been prepared according to the Electrical Engineering (EE) 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 Electrical Engineering (EE) 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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