Q21P - Mathematical Methods in Physical Sciences

Question

Evaluate each of the following in$\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$ form, and compare with a computer solution.$\mathit{c}\mathit{o}\mathit{s}\mathbf{\left(}\mathbf{i}\mathbf{l}\mathbf{n}\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{\right)}$

Step-by-Step Solution

Answer

The value of$\cos \left(i\ln (-1)\right)$ is -1.

Step 1: Given Information.

The given hyperbolic number is$\cos \left(i\ln (-1)\right)$

 

Step 2: Definition of Complex Number.

The numbers that are presented in the form of$\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$ where,x  is the real numbers and y  is an imaginary number, those numbers are referred to as called Complex numbers.  

Step 3: Find the value of. c o s ( i l n ( − 1 ) )

Use the identity  $\cos \left(iz\right)=\cosh \left(z\right)$

So,

$\begin{array}{c}\cos \left(i\ln (-1)\right)=\cosh \left[\ln (-1)\right]\\ =\cosh \left[\ln \left(1\right)+i(\pi \pm 2n\pi )\right]\\ =\cos (\pi \pm 2n\pi )\\ =-1\\ n=0,1,2,3,\mathrm{...}\end{array}$

 

Hence, the value of$\cos \left(i\ln (-1)\right)$ is -1.