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An analytic function of a complex variable z = x+iy (i = √-1) is defined as
f (z)= x2 − y2+i ψ ( x,y)
where ψ(x, y) is a real function. The value of the imaginary part of f(z) at z = (1 + i) is ___________ (round off to 2 decimal places).
f (z)= x2 − y2+i ψ ( x,y)
where ψ(x, y) is a real function. The value of the imaginary part of f(z) at z = (1 + i) is ___________ (round off to 2 decimal places).
Correct answer is '2'. Can you explain this answer?
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Verified Answer
An analytic function of a complex variable z = x+iy (i =√-1)is d...
f (z)= φ + i ψ is analytic
φ = x2 – y2
φx =2x = ψy φy = –ψx
φy =–2y = –ψx
ψx =2y ⇒ ψ = 2xy + C1
ψy =2x ⇒ ψ = 2xy + C2
Comparing ψ =2 xy + C
valid for all C put C = 0
ψ (1 + i) ⇒ (x = 1 y = 1)
∴ ψ =2
φ = x2 – y2
φx =2x = ψy φy = –ψx
φy =–2y = –ψx
ψx =2y ⇒ ψ = 2xy + C1
ψy =2x ⇒ ψ = 2xy + C2
Comparing ψ =2 xy + C
valid for all C put C = 0
ψ (1 + i) ⇒ (x = 1 y = 1)
∴ ψ =2
Most Upvoted Answer
An analytic function of a complex variable z = x+iy (i =√-1)is d...
The imaginary unit) is a function that can be expressed as a power series in z and its complex conjugate z̄. In other words, it can be written as:
f(z) = ∑(n=0 to ∞) [aₙ(z - z₀)ⁿ + bₙ(z - z₀)ⁿ̄]
where aₙ and bₙ are complex coefficients, z₀ is a complex constant, and z̄ is the complex conjugate of z.
Analytic functions have the property that they are differentiable at every point in their domain. This means that the derivative of an analytic function exists at every point in its domain and can be calculated using the limit definition of the derivative.
Analytic functions also satisfy the Cauchy-Riemann equations, which are a set of necessary and sufficient conditions for a function to be analytic. These equations relate the partial derivatives of the real and imaginary parts of the function.
Analytic functions have many important properties and are widely used in complex analysis and other areas of mathematics. They are particularly useful in solving problems involving complex integration and complex differential equations.
f(z) = ∑(n=0 to ∞) [aₙ(z - z₀)ⁿ + bₙ(z - z₀)ⁿ̄]
where aₙ and bₙ are complex coefficients, z₀ is a complex constant, and z̄ is the complex conjugate of z.
Analytic functions have the property that they are differentiable at every point in their domain. This means that the derivative of an analytic function exists at every point in its domain and can be calculated using the limit definition of the derivative.
Analytic functions also satisfy the Cauchy-Riemann equations, which are a set of necessary and sufficient conditions for a function to be analytic. These equations relate the partial derivatives of the real and imaginary parts of the function.
Analytic functions have many important properties and are widely used in complex analysis and other areas of mathematics. They are particularly useful in solving problems involving complex integration and complex differential equations.
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An analytic function of a complex variable z = x+iy (i =√-1)is defined asf (z)= x2 − y2+i ψ ( x,y)where ψ(x, y) is a real function. The value of the imaginary part of f(z) at z = (1 + i) is ___________ (round off to 2 decimal places).Correct answer is '2'. Can you explain this answer?
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An analytic function of a complex variable z = x+iy (i =√-1)is defined asf (z)= x2 − y2+i ψ ( x,y)where ψ(x, y) is a real function. The value of the imaginary part of f(z) at z = (1 + i) is ___________ (round off to 2 decimal places).Correct answer is '2'. 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 An analytic function of a complex variable z = x+iy (i =√-1)is defined asf (z)= x2 − y2+i ψ ( x,y)where ψ(x, y) is a real function. The value of the imaginary part of f(z) at z = (1 + i) is ___________ (round off to 2 decimal places).Correct answer is '2'. Can you explain this answer? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for An analytic function of a complex variable z = x+iy (i =√-1)is defined asf (z)= x2 − y2+i ψ ( x,y)where ψ(x, y) is a real function. The value of the imaginary part of f(z) at z = (1 + i) is ___________ (round off to 2 decimal places).Correct answer is '2'. Can you explain this answer?.
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