We are given the function $f(z) = (x^2 - 2xy + ay^2) + i(bx^2 - y^2 + 2xy)$, and we need to find constants $a$ and $b$ such that $f(z)$ is analytic.

Step 1: Understanding the function

The function $f(z)$ is expressed in terms of $x$ and $y$, where:

• $z = x + iy$ (so $x$ and $y$ are the real and imaginary parts of $z$).
• The real part of $f(z)$ is $u(x, y) = x^2 - 2xy + ay^2$.
• The imaginary part of $f(z)$ is $v(x, y) = bx^2 - y^2 + 2xy$.

Step 2: Applying the Cauchy-Riemann equations

For the function to be analytic, it must satisfy the Cauchy-Riemann equations:

1. $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$
2. $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$

First Cauchy-Riemann equation:

We start by calculating the partial derivatives.

• $u(x, y) = x^2 - 2xy + ay^2$ $\frac{\partial u}{\partial x} = 2x - 2y$

• $v(x, y) = bx^2 - y^2 + 2xy$ $\frac{\partial v}{\partial y} = 2xy - 2y + bx^2$

Equating them from the first Cauchy-Riemann equation: $2x - 2y = 2xy - 2y + bx^2$ Simplifying this: $2x = 2xy + bx^2$ Dividing both sides by 2: $x = xy + \frac{b}{2}x^2$ For this equation to hold true for all $x$ and $y$, we need $b = 1$.

Second Cauchy-Riemann equation:

Now calculate the other partial derivatives.

• $\frac{\partial u}{\partial y} = -2x + 2ay$

• $\frac{\partial v}{\partial x} = 2bx + 2y$

From the second Cauchy-Riemann equation: $-2x + 2ay = -2bx - 2y$ Substituting $b = 1$: $-2x + 2ay = -2x - 2y$ This simplifies to: $2ay = -2y$ For this to hold for all $y$, we must