[YMP/CV-05001] Elementary Functions --- Solved Examples

Solved Examples 

 Elementary Functions                                                                                                       Existence of Derivative w,r.t. \(z\) 

 NOTE: In this set we will take a look about existence of derivative w.r.t. complex variable $z$.

1. The functions $c^x$, $\sin x$, $\cos x$ are continuous and differentiable every where. WHY?\\ Simplest way to see is that for these functions derivative exists everywhere in the real line. Therefore they are continuous on the real line.
2. Consider the exponential of a complex variable $\exp(z)$. Its real and imaginary parts are given by \begin{align*} \exp(z) = & e^x(\cos y+i\sin y)\\ u(x,y) = & e^x\cos y,\qquad v(x,y)=e^x\sin y \end{align*}
3. Do the functions $u(x,y)$, and $v(x,y)$ satisfy the Cauchy Riemann equations? The answer is YES.
4. Find partial derivations $$ \frac{\partial u}{\partial x}~,~\frac{\partial V}{\partial x}~, \frac{\partial u}{\partial y}~,~\frac{\partial V}{\partial y} $$ Are these continuous functions of $x,y$?
5. The answers to $Q3$ and $Q4$ is YES. Therefore, $e^z$ is differentiable at all points.
6. The statement (5) implies that $\sin z$, $\cos z$, $\sinh z$, $\cosh z$ are all differentiable for all $z$.
7. $\tan z$, see $z$ are differentiable everywhere except where $\cos z=0$.
8. $\tanh z$, $\cosh z$ are differentiable everywhere except where $\cosh z=0$.
9. $\cot z$, ${\rm cosec} z$ are differentiable except where $\sin z $ is zero because $$ \cos z = \frac{\cos z}{\sin z}~,~{\rm cosec}z = \frac{1}{\sin z} $$
10. $\coth z$, ${\rm cosec\,h}z$ are differentiable everywhere except where $\sinh z=0$ because $$ \coth z = \frac{\cosh z}{\sinh z}~,~{\rm cosec\,h}z=\frac{1}{\sinh z} $$