[STR/CV-05001]-Summary of Points to be Remember

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Polynomial in $z$ is analytic everywhere.
Rational functions $P(z)/Q(z)$, where $P(z),Q(z)$ are polynomials, are analytic everywhere except where $Q(z)$ is zero.
$\exp(\lambda z)$, $\sin(\lambda z)\cos(\lambda z)$, $\sinh(\lambda z)\cosh(\lambda z)$ are also analytic everywhere.
other trigonometric and hyperbolic functions have singular points as given in the table. \begin{center}

Functions Singular points

$\tan z, \sec z$ $z=(2n+1)\pi/2$

$\cot z, {\rm cosec}\, z$ $z=n\pi$

$\tanh z, {\rm sech}\, z$ $z=(2n+1)i \pi/2$

$\coth z, {\rm cosech}\, z$ $z=n\pi i$
$\exp(\lambda z)$ does not become zero anywhere because $$ \exp(\lambda z)\exp(-\lambda z)=1 $$

4727:Diamond Point