If f(z) = (x² + ay²) + i bxy is a complex analytic function of z = x + iy, where $i=\sqrt{-1}$, then

Complex Analytic Function Explained

A complex function $f(z)=u(x,y)+iv(x,y)$ is considered an analytic function (or holomorphic function) in a domain if it is differentiable at every point in that domain. For a function to be analytic, its real part $u(x,y)$ and imaginary part $v(x,y)$ must satisfy the Cauchy-Riemann (CR) equations. These equations are fundamental in complex analysis for determining the analyticity of a complex function.

Cauchy-Riemann Equations for Analyticity

The Cauchy-Riemann equations state that for an analytic function $f(z)=u(x,y)+iv(x,y)$, the following conditions must hold:

- First Cauchy-Riemann Equation: $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$
- Second Cauchy-Riemann Equation: $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$

In this problem, we are given the complex analytic function $f(z)=(x^2+a{y}^2)+ibxy$. Our goal is to find the values of constants 'a' and 'b' that make this function analytic.

Identifying Real and Imaginary Parts

From the given function $f(z)=(x^2+a{y}^2)+ibxy$, we can identify its real part $u(x,y)$ and imaginary part $v(x,y)$:

- Real part: $u(x,y)=x^2+a{y}^2$
- Imaginary part: $v(x,y)=bxy$

Calculating Partial Derivatives

Next, we need to calculate the partial derivatives of $u$ and $v$ with respect to $x$ and $y$.

- Partial derivative of $u$ with respect to $x$: $$\frac{\partial u}{\partial x}=\frac{\partial }{\partial x}(x^2+a{y}^2)=2x$$
- Partial derivative of $u$ with respect to $y$: $$\frac{\partial u}{\partial y}=\frac{\partial }{\partial y}(x^2+a{y}^2)=2ay$$
- Partial derivative of $v$ with respect to $x$: $$\frac{\partial v}{\partial x}=\frac{\partial }{\partial x}(bxy)=by$$
- Partial derivative of $v$ with respect to $y$: $$\frac{\partial v}{\partial y}=\frac{\partial }{\partial y}(bxy)=bx$$

Applying Cauchy-Riemann Equations to Find a and b

Now, we apply the two Cauchy-Riemann equations using the partial derivatives we just calculated.

Applying the First Cauchy-Riemann Equation

The first equation is $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$.

$$2x=bx$$

For this equation to hold true for all values of $x$ in the domain, the coefficients of $x$ on both sides must be equal.

$$2=b$$

So, we find that the value of b = 2.

Applying the Second Cauchy-Riemann Equation

The second equation is $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$.

$$2ay=-(by)$$

Now, substitute the value of $b=2$ that we just found into this equation:

$$2ay=-(2y)$$

To satisfy this equation for all values of $y$, the coefficients of $y$ on both sides must be equal.

$$2a=-2$$

Divide both sides by 2:

$$a=\frac{-2}{2}$$

$$a=-1$$

So, we find that the value of a = -1.

Conclusion for a and b values

For the given complex function $f(z)=(x^2+a{y}^2)+ibxy$ to be an analytic function, the constants 'a' and 'b' must take the values:

- $a=-1$
- $b=2$

This corresponds to the option where a = -1 and b = 2.