1. Using Cauchy-Riemann equations of Cartesian form find the derivative (i.e f'(z)) of f(z) = z^2. 2. Show that the function f(z) = re^{i heta} has a derivative everywhere in its domain using the Cauchy-Riemann equations of polar form. 3. Show that u(x, y) = xe^x cos y - ye^x sin y is harmonic. 4. Find the harmonic conjugate v(x, y) of u(x, y) where u(x, y) = xe^x cos y - ye^x sin y . 5. Using Cauchy-Riemann equations determine whether f(z) = z^3 is analytic or not.

          1. Using Cauchy-Riemann equations of Cartesian form find the derivative (i.e f'(z)) of f(z) = z^2.

2. Show that the function f(z) = re^{i	heta} has a derivative everywhere in its domain using the Cauchy-Riemann equations of polar form.

3. Show that u(x, y) = xe^x cos y - ye^x sin y is harmonic.

4. Find the harmonic conjugate v(x, y) of u(x, y) where u(x, y) = xe^x cos y - ye^x sin y .

5. Using Cauchy-Riemann equations determine whether f(z) = z^3 is analytic or not.

1. Using Cauchy-Riemann equations of Cartesian form find the derivative (i.e f'(z)) of f(z) = z^2.

2. Show that the function f(z) = re^i heta has a derivative everywhere in its domain using the Cauchy-Riemann equations of polar form.

3. Show that u(x, y) = xe^x cos y - ye^x sin y is harmonic.

4. Find the harmonic conjugate v(x, y) of u(x, y) where u(x, y) = xe^x cos y - ye^x sin y .

5. Using Cauchy-Riemann equations determine whether f(z) = z^3 is analytic or not.

Added by Ashley P.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

01/12/2023

Step 1

We can do this by using the Cauchy-Riemann equations of Cartesian form. We find that the derivative of f(2) at (2,0) is given by: f'(2) = 2 - {(-1)^2} which is clearly analytic. Show more…

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1. Using Cauchy-Riemann equations of Cartesian form find the derivative (i.e f'(z)) of f(z) = z^2. 2. Show that the function f(z) = re^{i heta} has a derivative everywhere in its domain using the Cauchy-Riemann equations of polar form. 3. Show that u(x, y) = xe^x cos y - ye^x sin y is harmonic. 4. Find the harmonic conjugate v(x, y) of u(x, y) where u(x, y) = xe^x cos y - ye^x sin y. 5. Using Cauchy-Riemann equations determine whether f(z) = z^3 is analytic or not.