COMPLEX ANALYSIS please help!

Q: F. Prove the following: (10 pts each) 1.

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Q: dz The possible value for , where is any circle not passing through z = a, traversed z-a once in the…

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Q: Is the function eiz² holomorphic in C? Explain why. What are the Cauchy-Riemann equations for eiz?

A: Given function : f(z) = ei z2

Q: Evaluate | (6x² +2y²) dx + 4xy dy, where C is given by x = t. y = t,

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Q: (c) Show that (d/dz)z does not exist anywhere,

A: See the attachment

Q: Which identity of the following is true for the relation between the trigonometric and the…

A: By using the different relations between trigonometric and hyperbolic complex functions we can check…

Q: What is the exterior of the A. {2 € 0:1 21 >2} 3. { 26.0:12/51} annulus {2€ C:1 ≤|²|≤ 2}? c. { z €…

A: We have to solve given problems:

Q: Suppose (Zn)-1 CC and for each n € N+, let zn = xn+iyn, where an, yn E R. Suppose z = x+iy, with r,…

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Q: Find the Laurent expansion of that are valid for (a) |z| 2; f(z): = 1 (z+1)(2+2) 1 2+1 1 2+2

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Q: (A) h(x,y)= sinh r cos y+ icosh r sin y. O (B) h(r,y) = e" cos y + ie" sin y. O (C) h(r,y) = cos ry…

A: A function f is harmonic if ∂2f∂x2+∂2f∂y2=0

Q: [121² dz, where C' is the boundary of the triangle with vertices at th Evaluate points 0, 1, and i…

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Q: Q5: (a). Show that if h: IR2 IR and f= 2h3 + ih is an entire function, then h is a constant

A: An entire function satisfies cauchy reimann equation

Q: If Im{f'(x)} = 6x(2y-1) and f(0) = 3– 2i, f(1) = 6 - 5i. Find f(1+i). O (A) f(1+ i) = -6- 3i. O (B)…

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Q: Illustiate the reflection poinipal with the help of examples.

A: The principle of reflection is that when a ray of light is reflected from a smooth surface the angle…

Q: For each of the following functions, do the following: Find all its singularities in C For each…

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Q: Complex Analysis

A: to test the function u(x,y) for harmonic condition, and if so,find its harmonic conjugate v

Q: and its der Foo the Fin d function f(z) = | - K i+ 2+eZ %3D

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Q: Suppose f has a zero of order k at zo. Show that ‹ƒ′(²). f(z); ²0) = k. Res( (Hint: Express ƒ(z) =…

A: f has a zero of order k at z0 We have to prove Resf'zfz;z0=k

Q: Find Radius of converges to the series =0 -) zn

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Q: 1337 i+1` Compute ()

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Q: 4.) Let f(z) be an analytic function. Show that the following equality is hold. %3D

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Q: Let f (2) = 2² + 71m (2). If f(2)= u(x,y) + iv (x,y), find u(x,y) C. 2xy-y² A. x² + xy-y² D. xy-y²…

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Q: Find a parametrization for the line segment from 2₁ =1-i to Z₂ = 2i. A. 2 (t) = 1 -i + (−1+3i) t 0…

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Q: Show that U(,y) is Harmonic function and find its conjugate for U(x, y) = 2x – x³ + 3xy² -

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Q: Evaluate sin (716 + 2ei69 do. 2n Jo

A: Given integral is12π∫02πsin2π6+2eiθdθThe integral is evaluated as shown below

Q: Complex Analysis

A: For given function, First we will show the function is harmonic function as shown below :From the…

Q: Find the Fourier series of the given function f (t) = {" if - 1<t< O if 0<t<1 where period T = 2.

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Q: Which of the following represents the set D on the complex plane?

A: Given set: D=z∈ℂ:z-1+i&lt;2. We have to find the graph which represents D.

Q: Let a be a complex number 0. What is the absolute value of a/ā? O (A) 1. (B) 2. (C) 3. 4. O 000

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Q: Express the complex number equation below into exponential form. 2i (3 + i) (1 + iv3) = -8 2e# )…

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Q: True-False. If the assertion is true, quote a relevant theorem or reason, or give a proof; if false,…

A: (a) Consider the function fz=1sin1z. Observe that for any k∈ℤ\0, 1kπ is singularity of fz=1sin1z.…

Q: 1 dx 1) Evaluate the integral / G sinh x

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