1. Let z = x+iy with x, y Є R. Let f(z) = u(x, y) + iv(x, y) whereu(x, y), v(x, y): R² → R.(a) Suppose that f is complex differentiable. State the Cauchy-Riemann equationssatisfied by the functions u(x, y) and v(x,y).(b) State what it means for the function(2 mark)u(x, y): R² → Rto be a harmonic function.(3 marks)(c) Show that the function u(x, y) = 3x²y - y³ +2 is harmonic.(d) Find a harmonic conjugate of u(x, y).(6 marks)(9 marks)

(a) Cauchy-Riemann EquationsFor f(z) = u(x, y) + iv(x, y), the Cauchy-Riemann equations…

Answered: 1. Let z = x+iy with x, y Є R. Let f(z)…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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15) Let u = x³y - y³x a. Verify that the function u is harmonic in the whole complex plane b. Find the harmonic conjugate function v that ensures that ƒ (z) = u(x, y) + iv(x, y) is analytic.
Let f(2) = 22² – iz be a complex valued function defined on a complex plane. Writing z = x + iy, find real valued functions u(x, y) and v(x, y) such that f(2) = u(x, y) + iv(x, y). u(x, y) = 2x² + 2y² + x_and v(x, y) = 4xy – y u(x, y) = 2x° + 2y² – x and v(x, y) = 4.xy+ y u(x, y) = 2x? – 2y² + y and v(x, y) = 4xy – x u(x, y) = x² +y +x and v(x, y) = 2xy– y Home End PgUp PrtScn F7 F6 F8 F9
4. ) Find the directional derivative of the function f(x, y) = sin²(xy) at the point P (2,7) in the direction of the unit vector ū = −²¹+] √5
c) Given the scalar function (x, y, z) = x²e¹- y³ + cos2z. 3π 4 Find the directional derivative of p(x, y, z) at the point Q(1,2,) in the direction of a = 2 i-2j+k. Hence, obtain the direction and the value of maximum change of p(x, y, z) at the point Q.
Show that the function given by f (x, y) = x² + 3xy is differentiable at every point in the plane. By using the definition.
1. Obtain the directional derivative of: a.. f(x,y) = x²-4x³y² at the point (1,-2) in the direction of a unit vector whose angle with the semi-axis x is 1, u = cos i + sen j b. f(x,y) = x²-xy + 3y² at the point (-1,-2) in the direction of a unit vector whose angle with the semi-axis x is u = cos 0 i+sen j c. f(x,y) = x²sin y at the point (1.1) in the direction of a vector v = 3i-4j
Determine whether the complex functions given below satisfy the Cauchy-Riemann equations (have derivatives).
i) Find all values of z such that 25 = -1+ v3i. Plot your solutions on a complex plane. ii) For a complex function f(2) = u(x, y) + iv(x, y) with u and v real, the Cauchy-Riemann conditions state that f(2) is analytic if and only if dv du and ду du dx dy Let fi(2) and f2(2) be two analytic functions: prove that the function g(2) defined by g(2) = fi(2)f2(2) is also an analytic function. iii) Using the result of part (ii), or otherwise, determine whether the function f(z) = ze is analytic.
285. Find the scalar equation of the plane that passes through point P(-1, 2, 1) and is perpendicular to the line of intersection of planes x+y-z-2=0 and 2xy + 3z-1=0.
3. If F = 3xyi-y²j, evaluate [F. dr where C is the curve in the xy plane, y = 2x², from (0, 0) to (1, 2). [Ans: -2] 6
6. Let f(x, y) = ln(2x - y²). (a) (b) (c) Determine the gradient of f at the point (2,-1). Determine the equation of the plane tangent to the graph of f at the point (3,2). Find all unit vectors u such that Duf at (3, 2) is 0.
Express dw/ðu and əw/dv as functions of u and v both by using the Chain Rule and expressing w directly in terms of u and v before differentiating. w = In(x? + y? + z²), x= e" sinu, y=e" cos u, z= ue ": (u,v) = (-2,0)

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