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show that the function v=-y/(x^2+y^2), is harmonic and determine its conjugate function

To show that the function \( v = \frac{-y}{x^2 + y^2} \) is harmonic, we need to verify that it satisfies the **Laplace equation**:

\[
\Delta v = \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} = 0
\]

### Step 1: Compute the partial derivatives of \( v \)
We begin by computing the first and second partial derivatives of \( v \) with respect to \( x \) and \( y \).

1. **First partial derivative of \( v \) with respect to \( x \):**

\[
v = \frac{-y}{x^2 + y^2}
\]
Using the quotient rule, we get:
\[
\frac{\partial v}{\partial x} = \frac{(x^2 + y^2)(0) - (-y)(2x)}{(x^2 + y^2)^2} = \frac{2xy}{(x^2 + y^2)^2}
\]

2. **First partial derivative of \( v \) with respect to \( y \):**

Similarly, using the quotient rule:
\[
\frac{\partial v}{\partial y} = \frac{(x^2 + y^2)(-1) - (-y)(2y)}{(x^2 + y^2)^2} = \frac{-(x^2 - y^2)}{(x^2 + y^2)^2}
\]

### Step 2: Compute the second derivatives of \( v \)
Now, compute the second partial derivatives.

1. **Second partial derivative of \( v \) with respect to \( x \):**
Differentiate \( \frac{\partial v}{\partial x} = \frac{2xy}{(x^2 + y^2)^2} \) with respect to \( x \) again, using the quotient rule:
\[
\frac{\partial^2 v}{\partial x^2} = \frac{2y[(x^2 + y^2)^2] - 2xy(2(x^2 + y^2)2x)}{(x^2 + y^2)^4}
\]

Simplifying this expression is tedious, but it results in:

\[
\frac{\partial^2 v}{\partial x^2} = \frac{-2y(x^2 - y^2)}{(x^2 + y^2)^3}
\]

2. **Second partial derivative of \( v \) with respect to \( y \):**
Differentiate \( \frac{\partial v}{\partial y} = \frac{-(x^2 - y^2)}{(x^2 + y^2)^2} \) with respect to \( y \):
\[
\frac{\partial^2 v}{\partial y^2} = \frac{-2y(x^2 - y^2) + 2y(x^2 + y^2)^2}{(x

show that the function v=-y/(x^2+y^2), is harmonic and determine its conjugate function

This solution verifies that the function v = -y / (x^2 + y^2) is harmonic by using the Laplace equation, and includes detailed steps to find its conjugate function. Explore key concepts such as partial derivatives and complex analysis. Suitable for college-level students.

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