1. Let u be a smooth harmonic function inside the unit ball B(0, 1) in R? so that |u| < 1. Prove that |Vu(0)| <

Q: 2. Consider the PDE u; - t ux+tu=0 for - o <x< ∞ and 0 <t< ∞ with u (0, x) = sine x. Determine u (t,…

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Q: he characteristic curves for the IVP u+4 u,+2 u 0; (x, 0) = sin x; - 00 C are O a. T= X-t . T=X- 8…

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Q: The function f(z) =e² is: u(x.y)=e*cos(y) o u(x,y) =e*sin(y) ¤ contiuous in СО discontiuous in CO

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Q: Let f (z)=Z. Find Jf (z )dz from z = 0 to z =4+2i alongthe curve C given z 2i to z =2i.

A: We have to solve the integral ∫Cf(z)dz The given function f(z)=z¯. The curve is given from z=0 to…

Q: Let f(x, y, z) = ln () Vxy – sin (cos (zy)) , evaluate ху %3D дх ду

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Q: 4) Let F(x, y) = yi+xj. a) Show that F is conservative and find a function f such that F = = Vf. b)…

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Q: Find the singularity of the function f(z)=(e^z-1)/z at z=0.

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Q: B) Use Green's Theorem to evaluate F dr, where C is the triangle with vertices (0,0) (3,0) and (3,…

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Q: Find the Wronskian of the functions f(t) = et sin(t) and g(t) = et cos(t). NOTE: Your answer must be…

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Q: If f is a smooth, nonnegative function on [a, b], then the surface area S of the surface of…

A: Given that, f is a smooth i.e. its a differential curve on a,b and, It is asked to determine the…

Q: Consider the function f defined on [0, 1] such that f(0) = 1, ƒ(1) = 3, f(1/5) = f(3/5) = S, f(x)dx…

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Q: Let F(x, y, z) = (e" cos y, –e* sin y, 2). Evaluate the integral | F. dr, where C is the curve…

A: Given that, Fx, y, z=excosy, -exsiny, 2 Here, M=excosy, N=-exsiny, P=2. Differentiating M, N and P,…

Q: 2 Let f (x) = x on [0, 1] Calculate x dx and x dx by dissecting O, I] oaual narts and hence show…

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Q: Prove tht the fallowing ralt U(1-k)-2U(t) + U(t + k) U" (t) =- with an enor of ortler k?, which is…

A: The objective is to prove the central difference approximation to the second derivative of Ut…

Q: 10. Evaluate the integral cz(z-1) where C is a simple closed curve in the positive direction that…

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Q: Use Green's Theorem to evaluate the integral (y - x)dx + (2x - y)dy for the given path. C: x = 2…

A: Definition used -                                According to the green theorem -…

Q: (2) Find a smooth curve a in GL(2, R) passing through the identity such that a'(0) 1 0 Justify your…

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Q: Find a potential function for F or determine that F is not conservative. F = (cos(z), 16y, -x…

A: The vector field given in the question is: F = &lt;cos(z), 16y, -xsin(z)&gt; Hence, F1 = cos(z), F2…

Q: 1. Use the Cauchy-Riemann conditions to find out whether the following functions are analytic. a.…

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Q: Show that u(x, y) is harmonic in some domain. u(x, y) = 2x – x³ + 3xy?

A: since Uxx+Uyy=0, then U is harmonic

Q: Use the Cauchy-Riemann Equations to show that f(z) = z Im(z) is only differentiable at z =0 and find…

A: Cauchy-Riemann Equations: Let f=u+iv be a complex valued function , then the Cauchy-Riemann…

Q: e? The singular points of f (z) = that lies inside |z| are z(z²-1) 2 a. 1, –1 Ob. 0, –1 Ос. 0, 1 d.

A: We have to find the singular points of f(z) = ( ez / z(z2 - 1) ). that lies inside | z | = 1/2 are

Q: Let z = = g(x, y) = e* cos y +y In(x – 1).

A: Let we havez=g(x,y)=excosy+yln(x-1)If x=s-t and y=s2-t2

Q: (a) Find a harmonic function u on the upper half-plane that has the following boundary values. (1…

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Q: 4- Calculate the Laplacian of the following functions: (a) T e* sin4y cos 3z. (b) V = x° x^+3xz y-…

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Q: Suppose that (z) is entire and that the harmonic function u(x, y) = Re[fz)] has an upper bound uo;…

A: As requested following is the answer for question 12.

Q: Let f:C→C be defined by exp(-z-4) z#0 f(z) =. z = 0 Does f satisfy the Cauchy- Riemann equations? Is…

A: It is given that fz=e-1z4     z≠00           z=0 where f:ℂ→ℂ. We have to check whether the function…

Q: Given the vecor-valued function R(t) = cosht i ln(t+1) j+et k, find lim h→0 R(0+ h) - R(0) h O…

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Q: If z is a function of r and y implicitly defined by dz Əz find and dr ду at (x, y, z) = (0, 2, -1).…

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Q: The function f(z) =e² is: O contiuous in C discontiuous in C Ou(x,y) =e*cos(y) Ou(x,y)=e*sin(y)

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Q: HW.8. Use farseval's theorem to find if f(t)=A e-alti Ans: A?

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Q: Prove that the function f(z2) defined by x*(1+ i) – y°(1 – 1) , (z ± 0) x² + y? f(z) = [0, (z = 0)…

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Q: 3. Find the work done by F over the curve in the direction of increasing t. = (2,x, y) = (sin t, cos…

A: Write the vector field F=zi+xj+yk The equation of the curve. rt=sinti+costj+tk Differentiate…

Q: Which one of the following functions is not differentiable at z ? O (A) f(2) = 2 + 2 +1. O (B) f(2)…

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Q: Compute the Wronskian of a set of functions {vı, y2} = {e* cos(/), e* sin(v¤)} cos (Væ), e* sin(va)}…

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Q: Let C be the closed, piecewise smooth curve formed by traveling in straight lines between the points…

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Q: Let L(z) be the linearization of f(x) = sin x + 1 at the point x = . Find L(1/2). O -1 pi/2-1…

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Q: ▼ 3.1. Use Cauchy-Riemann equations to prove that ƒ(z) = z² is differentiable at z = zo E C.

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Q: - Let f(z) = cosh z. (1) Find the image of S := {z = x+iye C: x < 0,0 < y < } under f.

A: Follow the steps.

Q: Which one of the following functions is not differentiable at z O (A) ((z) = 2 +2 +1. O (B) f(2) = e…

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Q: he tangent plane to the function f (x, y) = cosh x + cos(ny) + e*+1)y at the point (0, 1, f(0, 1))…

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Q: (b) Show that the function f(2) =u +iv, where S(2) = **(l+1) – y'(1-1) x² + y² = 0, z = 0 Satisfies…

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Q: At what points (x, y, z) in space are the functions in continuous? h(x, y, z) = xy sin 1/z

A: Given function is  h(x, y, z) = xy sin 1z  We  have no additional conditions for x y part for any x…

Q: Solve the problem with Neumann oundary conditions, - Uxx = 0, x, t > 0 u(x, 0) = f(x) ux (0, t) = 0…

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Q: Let U (x, y) = e-(y sin x Find a real-valued function V (x, y) such that f(z) = U + iV is analytic.…

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Q: (a) Show that the limit x*ycosy lim (x.y)-(0,0) does not exist. (b) If u = x cos y+ye", wherex=s+2t…

A: The given function is f(x, y)=x2+y23 at point (0, 1) and direction u→=35, -45. The direction…

Q: 44. Let f(x, y) = g(u), where u = x² + y² and g(u) is differentiable. Prove that 2 4u (dg Әх du

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Q: %3D a) Find fyxyx for f(x,y) = In(xy) – cos (xy) at (1,t).

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Q: 1. For a function f (z) (with =x + iy) consisting of f (z) = u (x, y) + iv (x, y) Investigate…

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Q: Consider the function f defined on [0, 1] such that / (0) = f(1) = 1.f(1/5) = {(3/5) - 2, and /(2/5)…

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