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

See similar textbooks

Similar questions

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.
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.
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).
Could you explain how to show this in detail?
3. Let f(x, y) = In 2x + 4y, find the directional derivative of f(x, y) in the direction of at the point (-1, 1).
Let u(x.y) = e" (xcosy-y siny) (a) Show that u(x,y) is a harmonic function. (b) If v(x,y) is a harmonic Conjugate of u, find v(x,y). (c) Use the answers in part (a) and (b) to find an analytic function f(z) that satisfies. Use Cauchy-Riemann theorem to show that f'(z) = e* (z+1) (d)
Please don't provide handwritten solution ...
3. Calculate the directional derivative of the function f : R³ → R, defined by f(x, y, z) = y²+xz at the point p = (1,2, 2) and in the direction of the vector = (2, -1, 2)
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
1. Consider the vector-valued function F(t) = sin(2t) t (2, 1, -2), a) Find the domain of F. b) Show that F is continuous at t = 0. ; et, t 1-√1+t, t #0 t = 0. = (1/2/3/17, -2). 2. Consider the vector-valued function R(t) with R(1) = (1,2,−1) and R (t) = (1 a) Find '(1) if (t) = (-2, -3t, t²) × R'(t). b) Find the equation of the normal plane to the graph of Ả at t = 1. c) Find the arc length of the graph of R from the point at t = 1 to the point at t = 3. d) Find the curvature of the graph of R at t = 1.

Recommended textbooks for you

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Mathematics For Machine Technology

Advanced Math

ISBN:

9781337798310

Author:

Peterson, John.

Publisher:

Cengage Learning,

Basic Technical Mathematics

Advanced Math

ISBN:

9780134437705

Author:

Washington

Publisher:

PEARSON

Topology

Advanced Math

ISBN:

9780134689517

Author:

Munkres, James R.

Publisher:

Pearson,

SEE MORE TEXTBOOKS