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Complex Analysis : Analytic and Harmonic Functions

Study concepts, example questions & explanations for Complex Analysis

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Example Questions

Example Question #1 : Analytic And Harmonic Functions

Find a Harmonic Conjugate v(x,y) of u(x,y) = 2x-x^3+3xy^2

Possible Answers:

v(x,y) = 2-3x^2+y^3

v(x,y) = 2y-3x^2y+y^3

v(x,y) = 2x-x^3+y^3

v(x,y) = 2y-x^3y+3xy^2

v(x,y) = 2y-x^2y + y^3

Correct answer:

v(x,y) = 2y-3x^2y+y^3

Explanation:

v(x,y) is said to be a harmonic conjugate of u(x,y) if their are both harmonic in their domain and their first order partial derivatives satisfy the Cauchy-Riemann Equations. Computing the partial derivatives

u_x(x,y) = 2-3x^2+3y^2 = v_y(x,y)

v(x,y) = 2y-3x^2y+y^3 + C

where is any arbitrary constant.

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Example Question #2 : Analytic And Harmonic Functions

Given $f(z) = z - \bar{z}$ , where does f'(z) exist?

Possible Answers:

x = 0

The Entire Complex Plane

z=0

Nowhere

y = 0

Correct answer:

Nowhere

Explanation:

Rewriting f(z) in real and complex components, we have that

$f(z) = z- \bar{z} = (x+yi) - (x-yi) = 2yi$

So this implies that f(z) = f(x + yi) = u(x,y) + v(x,y)i

where $u(x,y) = 0 \text{ and } v(x,y) = 2y$

Therefore, checking the Cauchy-Riemann Equations, we have that

$u_x(x,y) = 0 \neq v_y(x,y) = 2$

So the Cauchy-Riemann equations are never satisfied on the entire complex plane, so f(z) is differentiable nowhere.

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Complex Analysis : Analytic and Harmonic Functions

Study concepts, example questions & explanations for Complex Analysis

All Complex Analysis Resources

Example Questions

Example Question #1 : Analytic And Harmonic Functions

Example Question #2 : Analytic And Harmonic Functions

All Complex Analysis Resources

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