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For an analytic function f (x iy) = u (x, y) iv (x, y), u is given by u = 4xy+ x. The expression for V considering K to be a constant
- a)y2 – 2y – 4x+ K
- b)x2+ y+ 4x
- c)2y2 + y – 2x2 + K
- d)- 4xy+ K
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
For an analytic function f (x iy) = u (x, y) iv (x, y), u is given by ...
Since f is analytic, it will satisfy Cauchy. Riemann equations
Most Upvoted Answer
For an analytic function f (x iy) = u (x, y) iv (x, y), u is given by ...
Understanding Analytic Functions
Analytic functions are functions that are complex differentiable in a neighborhood of every point in their domain. For a function f(z) = u(x, y) + iv(x, y) to be analytic, the real part u and the imaginary part v must satisfy the Cauchy-Riemann equations.
Cauchy-Riemann Equations
Given u(x, y) = 4xy + x, we need to find v(x, y) such that:
- ∂u/∂x = ∂v/∂y
- ∂u/∂y = -∂v/∂x
Calculating Partial Derivatives
1. Calculate ∂u/∂x:
- ∂u/∂x = 4y + 1
2. Calculate ∂u/∂y:
- ∂u/∂y = 4x
Using Cauchy-Riemann Equations
From the Cauchy-Riemann equations:
- ∂v/∂y = 4y + 1
- -∂v/∂x = 4x
Now, we can integrate these equations to find v.
Finding v(x, y)
1. Integrate ∂v/∂y:
- v = ∫(4y + 1) dy = 2y^2 + y + g(x)
2. Integrate -∂v/∂x:
- -∂v/∂x = 4x implies v = -2x^2 + h(y)
Here, g(x) and h(y) are functions of x and y only.
Combining Results
To combine results, we equate both expressions for v. Noting that any constant can be represented as K, we arrive at:
v = 2y^2 + y - 2x^2 + K
This matches option C.
Conclusion
Thus, the expression for V considering K to be a constant is:
Option C: 2y^2 + y - 2x^2 + K
Analytic functions are functions that are complex differentiable in a neighborhood of every point in their domain. For a function f(z) = u(x, y) + iv(x, y) to be analytic, the real part u and the imaginary part v must satisfy the Cauchy-Riemann equations.
Cauchy-Riemann Equations
Given u(x, y) = 4xy + x, we need to find v(x, y) such that:
- ∂u/∂x = ∂v/∂y
- ∂u/∂y = -∂v/∂x
Calculating Partial Derivatives
1. Calculate ∂u/∂x:
- ∂u/∂x = 4y + 1
2. Calculate ∂u/∂y:
- ∂u/∂y = 4x
Using Cauchy-Riemann Equations
From the Cauchy-Riemann equations:
- ∂v/∂y = 4y + 1
- -∂v/∂x = 4x
Now, we can integrate these equations to find v.
Finding v(x, y)
1. Integrate ∂v/∂y:
- v = ∫(4y + 1) dy = 2y^2 + y + g(x)
2. Integrate -∂v/∂x:
- -∂v/∂x = 4x implies v = -2x^2 + h(y)
Here, g(x) and h(y) are functions of x and y only.
Combining Results
To combine results, we equate both expressions for v. Noting that any constant can be represented as K, we arrive at:
v = 2y^2 + y - 2x^2 + K
This matches option C.
Conclusion
Thus, the expression for V considering K to be a constant is:
Option C: 2y^2 + y - 2x^2 + K
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For an analytic function f (x iy) = u (x, y) iv (x, y), u is given by u = 4xy+ x. The expression for V considering K to be a constanta)y2 – 2y – 4x+ Kb)x2+ y+ 4xc)2y2+ y – 2x2 + Kd)- 4xy+ KCorrect answer is option 'C'. Can you explain this answer?
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For an analytic function f (x iy) = u (x, y) iv (x, y), u is given by u = 4xy+ x. The expression for V considering K to be a constanta)y2 – 2y – 4x+ Kb)x2+ y+ 4xc)2y2+ y – 2x2 + Kd)- 4xy+ KCorrect answer is option 'C'. Can you explain this answer? for Mechanical Engineering 2025 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about For an analytic function f (x iy) = u (x, y) iv (x, y), u is given by u = 4xy+ x. The expression for V considering K to be a constanta)y2 – 2y – 4x+ Kb)x2+ y+ 4xc)2y2+ y – 2x2 + Kd)- 4xy+ KCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For an analytic function f (x iy) = u (x, y) iv (x, y), u is given by u = 4xy+ x. The expression for V considering K to be a constanta)y2 – 2y – 4x+ Kb)x2+ y+ 4xc)2y2+ y – 2x2 + Kd)- 4xy+ KCorrect answer is option 'C'. Can you explain this answer?.
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