Decompose a fraction into a sum of fractions (with complex variable)

Question

Here is equality:

$\frac{1}{z\ln4+(z\ln4)^2/2+O(z^3)}=\frac{1}{z\ln4}-\frac{1}{2}+O(z)$

I can't understand why is it correct?

Answer 1

Hint. One may recall that, as $u \to 0$, $$ \require{cancel}\frac1{1+u}=1-u+O(u^2) $$ giving, as $z \to 0$, $$ \begin{align} \frac{1}{z\ln4+(z\ln4)^2/2+O(z^3)}&=\frac{1}{z\ln4}\cdot\frac{1}{1+\underbrace{\frac{\ln4}2\cdot z+O(z^2)}_{\color{red}{u \,\text{with}\, u\to 0}}} \\\\&=\frac{1}{z\ln4}\cdot\left[1-\left(\frac{\ln4}2\cdot z+O(z^2)\right)+O(z^2)\right] \\\\&=\frac{1}{z\ln4}-\frac{1}{\cancel{z}\cancel{\ln4}}\cdot\frac{\cancel{\ln4}}2\cdot \cancel{z}+O(z) \\\\&=\frac{1}{z\ln4}-\frac12+O(z). \end{align} $$