How do I find the Laurent expansion of a function containing the principal branch cut of the nth root?
Example:
$\displaystyle f(z)=-iz\cdot\mathrm{pv}\sqrt[4]{1-\frac{1}{z^{4}}}$
How do I find the Laurent expansion of a function containing the principal branch cut of the nth root?
Example:
$\displaystyle f(z)=-iz\cdot\mathrm{pv}\sqrt[4]{1-\frac{1}{z^{4}}}$
Remembering the binomial series expansion...
$\displaystyle (1+x)^{\alpha}= \sum_{k=0}^{\infty} \frac{\alpha\cdot (\alpha-1)\dots (\alpha-k+1)}{k!}\cdot x^{k}$
... for $\displaystyle x=-\frac{1}{z^{4}}$ and $\displaystyle \alpha=\frac{1}{4}$ we have...
$\displaystyle (1-\frac{1}{z^{4}})^{\frac{1}{4}}= 1 - \frac{1}{4}\cdot z^{-4} - \frac{3}{4\cdot 4\cdot 2!}\cdot z^{-8} - \frac{3\cdot 7}{4\cdot 4\cdot 4\cdot 3!}\cdot z^{-12} + \dots$
... and then...
$\displaystyle f(z)= -i\cdot z \cdot (1-\frac{1}{z^{4}})^{\frac{1}{4}}= -i\cdot (z - \frac{1}{4}\cdot z^{-3} - \frac{3}{4\cdot 4\cdot 2!}\cdot z^{-7} - \frac{3\cdot 7}{4\cdot 4\cdot 4\cdot 3!}\cdot z^{-11} + \dots) $
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$