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The function f (z ) of complex variable z = x + iy , where i = −1 , is given as f (z ) = (x3 – 3xy2) + iv (x, y ). For this function to be analytic, v (x, y ) should be
- a)(3x2y – y3) + constant
- b)(3x2y2 – y3) + constant
- c)(x2 – 3x2y) + constant
- d)(3xy2 – y3) + constant
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
The function f (z ) of complex variable z = x + iy , where i = −...
f (z)= u + iv
u = x3 – 3 xy2, v = v (x, y )
For f (z ) to be Analytical,
ux =3x2 – 3y2 = vy
uy =–6 xy = –vx
vx =6 xy by integrating w.r.t x ⇒ v = 3x2y + C1
vy =3x2 – 3y2 by integrating w.r.t y ⇒ v = 3x2y – y3 + C2
v =(3x2y – y3) + constant (C1 = –y3)
u = x3 – 3 xy2, v = v (x, y )
For f (z ) to be Analytical,
ux =3x2 – 3y2 = vy
uy =–6 xy = –vx
vx =6 xy by integrating w.r.t x ⇒ v = 3x2y + C1
vy =3x2 – 3y2 by integrating w.r.t y ⇒ v = 3x2y – y3 + C2
v =(3x2y – y3) + constant (C1 = –y3)
Most Upvoted Answer
The function f (z ) of complex variable z = x + iy , where i = −...
The function f(z) of complex variable z = x + iy, where i = √(-1), is a function that maps a complex number z to another complex number. It can be written as f(z) = u + iv, where u and v are real-valued functions of x and y.
For example, let's say f(z) = z^2. In this case, we have f(z) = (x + iy)^2 = x^2 + 2ixy - y^2. So, u(x, y) = x^2 - y^2 and v(x, y) = 2xy.
Another example could be f(z) = e^z. In this case, we have f(z) = e^(x + iy) = e^x * e^(iy) = e^x * (cos(y) + i*sin(y)). So, u(x, y) = e^x * cos(y) and v(x, y) = e^x * sin(y).
The function f(z) can have various other forms depending on the specific function being considered. It can be a polynomial, exponential, logarithmic, trigonometric, or any other type of function. The specific form of f(z) will determine the behavior and properties of the function.
For example, let's say f(z) = z^2. In this case, we have f(z) = (x + iy)^2 = x^2 + 2ixy - y^2. So, u(x, y) = x^2 - y^2 and v(x, y) = 2xy.
Another example could be f(z) = e^z. In this case, we have f(z) = e^(x + iy) = e^x * e^(iy) = e^x * (cos(y) + i*sin(y)). So, u(x, y) = e^x * cos(y) and v(x, y) = e^x * sin(y).
The function f(z) can have various other forms depending on the specific function being considered. It can be a polynomial, exponential, logarithmic, trigonometric, or any other type of function. The specific form of f(z) will determine the behavior and properties of the function.
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The function f (z ) of complex variable z = x + iy , where i = −1 , is given as f (z ) = (x3 – 3xy2) + iv (x, y ). For this function to be analytic, v (x, y ) should bea)(3x2y – y3) + constantb)(3x2y2 – y3) + constantc)(x2 – 3x2y) + constantd)(3xy2 – y3) + constantCorrect answer is option 'A'. 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) + iv (x, y ). For this function to be analytic, v (x, y ) should bea)(3x2y – y3) + constantb)(3x2y2 – y3) + constantc)(x2 – 3x2y) + constantd)(3xy2 – y3) + constantCorrect answer is option 'A'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about The function f (z ) of complex variable z = x + iy , where i = −1 , is given as f (z ) = (x3 – 3xy2) + iv (x, y ). For this function to be analytic, v (x, y ) should bea)(3x2y – y3) + constantb)(3x2y2 – y3) + constantc)(x2 – 3x2y) + constantd)(3xy2 – y3) + constantCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for GATE 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) + iv (x, y ). For this function to be analytic, v (x, y ) should bea)(3x2y – y3) + constantb)(3x2y2 – y3) + constantc)(x2 – 3x2y) + constantd)(3xy2 – y3) + constantCorrect answer is option 'A'. Can you explain this answer?.
The function f (z ) of complex variable z = x + iy , where i = −1 , is given as f (z ) = (x3 – 3xy2) + iv (x, y ). For this function to be analytic, v (x, y ) should bea)(3x2y – y3) + constantb)(3x2y2 – y3) + constantc)(x2 – 3x2y) + constantd)(3xy2 – y3) + constantCorrect answer is option 'A'. Can you explain this answer? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about The function f (z ) of complex variable z = x + iy , where i = −1 , is given as f (z ) = (x3 – 3xy2) + iv (x, y ). For this function to be analytic, v (x, y ) should bea)(3x2y – y3) + constantb)(3x2y2 – y3) + constantc)(x2 – 3x2y) + constantd)(3xy2 – y3) + constantCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for GATE 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) + iv (x, y ). For this function to be analytic, v (x, y ) should bea)(3x2y – y3) + constantb)(3x2y2 – y3) + constantc)(x2 – 3x2y) + constantd)(3xy2 – y3) + constantCorrect answer is option 'A'. Can you explain this answer?.
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ample number of questions to practice The function f (z ) of complex variable z = x + iy , where i = −1 , is given as f (z ) = (x3 – 3xy2) + iv (x, y ). For this function to be analytic, v (x, y ) should bea)(3x2y – y3) + constantb)(3x2y2 – y3) + constantc)(x2 – 3x2y) + constantd)(3xy2 – y3) + constantCorrect answer is option 'A'. Can you explain this answer? tests, examples and also practice GATE tests.
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