From Example Problems
Jump to: navigation, search

Find the residues of f(z)\, at all its isolated singular points and at infinity (if infinity is not a limit point of singular points), where f(z)\, is given by {\frac  {e^{{iz}}}{{\sqrt  {z}}}}\,

This problem is currently in dispute. A reputable source thinks the given function is not representable by a Laurent series and so the residue is not defined. Investigation is proceeding.

Note that

{\frac  {e^{{iz}}}{{\sqrt  {z}}}}={\frac  {{\sqrt  {z}}e^{{iz}}}{z}}\,


Residue at z=0\, =\lim _{{z\to 0}}(z-0)f(z)\,
=\lim _{{z\to 0}}{\sqrt  {z}}e^{{iz}}
={\sqrt  {0}}e^{{i0}}

Main Page : Complex Variables : Residues