[PTR/CV-05001]-Points to Remember

                                                                                                                            Trigonometric and hyperbolic functions

Singular points and analytic property

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

   Functions	Singular points
   $\tan z, \sec z$	$z=(2n+1)\pi/2$
   $\cot z, {\rm cosec}\, z$	$z=n\pi$
   $\tanh z, {\rm sech}\, z$	$z=(2n+1)i \pi/2$
   $\coth z, {\rm cosech}\, z$	$z=n\pi i$

    

5. $\exp(\lambda z)$ does not become zero anywhere because $$ \exp(\lambda z)\exp(-\lambda z)=1 $$ So, for example, \(f(z)= \frac{\sin z}{\exp(-z)}\) is analytic everywhere.