Analytic and Harmonic Functions - Complex Analysis

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Question

Find a Harmonic Conjugate of

Answer

is said to be a harmonic conjugate of if their are both harmonic in their domain and their first order partial derivatives satisfy the Cauchy-Riemann Equations. Computing the partial derivatives

where is any arbitrary constant.

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Question

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

Answer

Rewriting in real and complex components, we have that

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

So this implies that

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

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

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

Compare your answer with the correct one above

Question

Find a Harmonic Conjugate of

Answer

is said to be a harmonic conjugate of if their are both harmonic in their domain and their first order partial derivatives satisfy the Cauchy-Riemann Equations. Computing the partial derivatives

where is any arbitrary constant.

Compare your answer with the correct one above

Question

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

Answer

Rewriting in real and complex components, we have that

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

So this implies that

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

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

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

Compare your answer with the correct one above

Question

Find a Harmonic Conjugate of

Answer

is said to be a harmonic conjugate of if their are both harmonic in their domain and their first order partial derivatives satisfy the Cauchy-Riemann Equations. Computing the partial derivatives

where is any arbitrary constant.

Compare your answer with the correct one above

Question

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

Answer

Rewriting in real and complex components, we have that

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

So this implies that

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

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

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

Compare your answer with the correct one above

Question

Find a Harmonic Conjugate of

Answer

is said to be a harmonic conjugate of if their are both harmonic in their domain and their first order partial derivatives satisfy the Cauchy-Riemann Equations. Computing the partial derivatives

where is any arbitrary constant.

Compare your answer with the correct one above

Question

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

Answer

Rewriting in real and complex components, we have that

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

So this implies that

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

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

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

Compare your answer with the correct one above

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Analytic and Harmonic Functions - Complex Analysis

Find a Harmonic Conjugate of

is said to be a harmonic conjugate of if their are both harmonic in their domain and their first order partial derivatives satisfy the Cauchy-Riemann Equations. Computing the partial derivatives where is any arbitrary constant.

Given , where does exist?

Rewriting in real and complex components, we have that So this implies that where Therefore, checking the Cauchy-Riemann Equations, we have that So the Cauchy-Riemann equations are never satisfied on the entire complex plane, so is differentiable nowhere.

Find a Harmonic Conjugate of

is said to be a harmonic conjugate of if their are both harmonic in their domain and their first order partial derivatives satisfy the Cauchy-Riemann Equations. Computing the partial derivatives where is any arbitrary constant.

Given , where does exist?

Rewriting in real and complex components, we have that So this implies that where Therefore, checking the Cauchy-Riemann Equations, we have that So the Cauchy-Riemann equations are never satisfied on the entire complex plane, so is differentiable nowhere.

Find a Harmonic Conjugate of

is said to be a harmonic conjugate of if their are both harmonic in their domain and their first order partial derivatives satisfy the Cauchy-Riemann Equations. Computing the partial derivatives where is any arbitrary constant.

Given , where does exist?

Rewriting in real and complex components, we have that So this implies that where Therefore, checking the Cauchy-Riemann Equations, we have that So the Cauchy-Riemann equations are never satisfied on the entire complex plane, so is differentiable nowhere.

Find a Harmonic Conjugate of

is said to be a harmonic conjugate of if their are both harmonic in their domain and their first order partial derivatives satisfy the Cauchy-Riemann Equations. Computing the partial derivatives where is any arbitrary constant.

Given , where does exist?

Rewriting in real and complex components, we have that So this implies that where Therefore, checking the Cauchy-Riemann Equations, we have that So the Cauchy-Riemann equations are never satisfied on the entire complex plane, so is differentiable nowhere.

is said to be a harmonic conjugate of if their are both harmonic in their domain and their first order partial derivatives satisfy the Cauchy-Riemann Equations. Computing the partial derivatives

where is any arbitrary constant.

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

is said to be a harmonic conjugate of if their are both harmonic in their domain and their first order partial derivatives satisfy the Cauchy-Riemann Equations. Computing the partial derivatives

where is any arbitrary constant.

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

is said to be a harmonic conjugate of if their are both harmonic in their domain and their first order partial derivatives satisfy the Cauchy-Riemann Equations. Computing the partial derivatives

where is any arbitrary constant.

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

is said to be a harmonic conjugate of if their are both harmonic in their domain and their first order partial derivatives satisfy the Cauchy-Riemann Equations. Computing the partial derivatives

where is any arbitrary constant.

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