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A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x2 − 2y2 + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) is
- a)4xy − 2x2 + 2y2 + constant
- b)4y2 − 4xy + constant
- c)2x2 − 2y2 + xy + constant
- d)−4xy + 2y2 − 2x2 + constant
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
A harmonic function is analytic if it satisfies the Laplace equation. ...
In order to determine whether the function u(x, y) = 2x^2 is harmonic, we need to check if it satisfies the Laplace equation. The Laplace equation in two dimensions is given by:
∇^2u = ∂^2u/∂x^2 + ∂^2u/∂y^2 = 0
Let's calculate the second partial derivatives of u(x, y):
∂^2u/∂x^2 = 4
∂^2u/∂y^2 = 0
Now, let's plug these values into the Laplace equation:
∂^2u/∂x^2 + ∂^2u/∂y^2 = 4 + 0 = 4
Since the Laplace equation is not satisfied (it should be equal to 0), we can conclude that the function u(x, y) = 2x^2 is not harmonic.
∇^2u = ∂^2u/∂x^2 + ∂^2u/∂y^2 = 0
Let's calculate the second partial derivatives of u(x, y):
∂^2u/∂x^2 = 4
∂^2u/∂y^2 = 0
Now, let's plug these values into the Laplace equation:
∂^2u/∂x^2 + ∂^2u/∂y^2 = 4 + 0 = 4
Since the Laplace equation is not satisfied (it should be equal to 0), we can conclude that the function u(x, y) = 2x^2 is not harmonic.
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Community Answer
A harmonic function is analytic if it satisfies the Laplace equation. ...
Let w = u + iν be a function of complex variable.
Function of a complex variable is analytic, if it satisfies Cauchy-reimann equation;
Calculation:
Given:
Function of a complex variable is analytic, if it satisfies Cauchy-reimann equation;
Calculation:
Given:
u(x, y) = 2x2 – 2y2 + 4xy, ν(x, y) = ?
∂u/∂x = ∂ν/∂y
∂u/∂x = 4x + 4y = ∂ν/∂y
∂u/∂x = 4x + 4y = ∂ν/∂y
Integrating w.r.t y keeping x constant
ν(x, y) = 4xy + 2y2 + f(x)
∂v/∂x = 4y + f′(x)
∂u/∂y =−∂ν*∂x
∂u/∂y = −4y+4x
4y – 4x = 4y + f’(x)
f(x) = −4x2*2 + C = −2x2 + C
∴ ν(x, y) = 4xy + 2y2 – 2x2 + C
∂v/∂x = 4y + f′(x)
∂u/∂y =−∂ν*∂x
∂u/∂y = −4y+4x
4y – 4x = 4y + f’(x)
f(x) = −4x2*2 + C = −2x2 + C
∴ ν(x, y) = 4xy + 2y2 – 2x2 + C
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A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x2 − 2y2 + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) isa)4xy − 2x2 + 2y2 + constantb)4y2 − 4xy + constantc)2x2 − 2y2 + xy + constantd)−4xy + 2y2 − 2x2 + constantCorrect answer is option 'A'. Can you explain this answer?
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A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x2 − 2y2 + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) isa)4xy − 2x2 + 2y2 + constantb)4y2 − 4xy + constantc)2x2 − 2y2 + xy + constantd)−4xy + 2y2 − 2x2 + constantCorrect answer is option 'A'. 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 A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x2 − 2y2 + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) isa)4xy − 2x2 + 2y2 + constantb)4y2 − 4xy + constantc)2x2 − 2y2 + xy + constantd)−4xy + 2y2 − 2x2 + constantCorrect answer is option 'A'. 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 A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x2 − 2y2 + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) isa)4xy − 2x2 + 2y2 + constantb)4y2 − 4xy + constantc)2x2 − 2y2 + xy + constantd)−4xy + 2y2 − 2x2 + constantCorrect answer is option 'A'. Can you explain this answer?.
A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x2 − 2y2 + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) isa)4xy − 2x2 + 2y2 + constantb)4y2 − 4xy + constantc)2x2 − 2y2 + xy + constantd)−4xy + 2y2 − 2x2 + constantCorrect answer is option 'A'. 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 A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x2 − 2y2 + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) isa)4xy − 2x2 + 2y2 + constantb)4y2 − 4xy + constantc)2x2 − 2y2 + xy + constantd)−4xy + 2y2 − 2x2 + constantCorrect answer is option 'A'. 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 A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x2 − 2y2 + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) isa)4xy − 2x2 + 2y2 + constantb)4y2 − 4xy + constantc)2x2 − 2y2 + xy + constantd)−4xy + 2y2 − 2x2 + constantCorrect answer is option 'A'. Can you explain this answer?.
Solutions for A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x2 − 2y2 + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) isa)4xy − 2x2 + 2y2 + constantb)4y2 − 4xy + constantc)2x2 − 2y2 + xy + constantd)−4xy + 2y2 − 2x2 + constantCorrect answer is option 'A'. Can you explain this answer? in English & in Hindi are available as part of our courses for Electrical Engineering (EE).
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ample number of questions to practice A harmonic function is analytic if it satisfies the Laplace equation. If u(x, y) = 2x2 − 2y2 + 4xy is a harmonic function, then its conjugate harmonic function v(x, y) isa)4xy − 2x2 + 2y2 + constantb)4y2 − 4xy + constantc)2x2 − 2y2 + xy + constantd)−4xy + 2y2 − 2x2 + constantCorrect answer is option 'A'. Can you explain this answer? tests, examples and also practice Electrical Engineering (EE) tests.
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