Show that the function defined by
$$f(z) = \begin{cases}
(x^3)(y^4)(x+iy) / (x^6 + y^{10}) & \text { when } z \ne 0 \\
0 & \text{ when } z = 0
\end{cases}$$
is not analytic at the origin.
First I have verified that Cauchy-Riemann equations of the given function are satisfied at the origin.
Now I tried to check the differentiability at the origin. The function is said to be differentiable at the origin if $\lim\limits_{h\to0} \frac{ f(h) – f(0)}{ h}$ exists.
Let $h = a+ib$, after substituting $h$ in the above limit, I am stuck with
$$\lim_{ (a,b) \to(0,0) } \frac{a^3b^5}{a^6 + b^{10}}$$
I have tried to approach origin through $x$ and $y$ axis and also through $y=mx$ and $x=my$, but the value of the limit is coming out to be $0$. Please help me to solve this.
Best Answer
If the real (or the imaginary) part of $z$ is $0$, then $f(z)=0$. Therefore, if $f$ was differentiable at the origin, then $f'(0)=0$. On the other hand$$(\forall x\in\mathbb{R}\setminus\{0\}):\frac{f(x^4+x^3i)}{x^4+x^3i}=\frac1{1+x^6}$$and therefore if $f$ was differentiable at the origin, $f'(0)=1$. So $f$ is not differentiable at the origin.