Let $f:\mathbb{C} \to \mathbb{C}$ be a function defined by $f(x+iy) = x^2-y^2+2cxy$ , where $c$ is a complex constant.
In other words, $c=c_1+ic_2 \in \mathbb{C}$.
We also define $x,y,c_1,c_2 \in \mathbb{R}$.
Find the values of $c$ for which $f$ is analytic in $\mathbb{C}$.
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I began this question by expanding the constant term in the definition of the function:
$h(x+iy)=x^2+y^2+2c_1xy+i(2C_2xy)$.
From this I worked out the Cauchy Riemann equations by taking $u(x,y)=x^2+y^2+2c_1xy$ and $v(x,y)=2C_2xy$.
Hence, $\frac{\partial u}{\partial x}=2x+2c_1y$ , $\frac{\partial u}{\partial y}=-2y+2c_1x$ , $\frac{\partial v}{\partial y}=2c_2x$ , $\frac{\partial v}{\partial x}=2c_2y$.
Then, for $h$ to be analytic in $\mathbb{C}$, all of these partial derivatives must satisfy the Cauchy Riemann equations. It is here that I am stuck and am not sure how to work out the constants $c_1$ and $c_2$. Any help would be very much appreciated!
Best Answer
CR equations are $$ \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y} $$ $$ \frac{\partial v}{\partial x}=-\frac{\partial u}{\partial y}\;. $$ The former is $2x+2c_1y=2c_2x$, from which immediately you get $c_1=0, c_2=1$. These values satisfy automatically the latter.
Thus the only complex value of $c$ to have $f$ analytic is $c=i$.