Q15P - 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.${\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{i}}$

 

 

Step-by-Step Solution

Answer

The value of${(-1)}^{\sin i}$ is $e^{\left[-\sinh \left(1\right)(\pi \pm 2n\pi )\right]}$Where.$n=0,1,2,3,\mathrm{...}$

 

Step 1: Given Information.

The given expression is.${(-1)}^{\sin i}$

Step 2: Definition of Complex Number and Hyperbolic functions.

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

The basic hyperbolic functions are:

1. Hyperbolic sine(sinh)
2.  Hyperbolic cosine(cosh)
3. Hyperbolic tangent(tanh)

 

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

It is known that  $\sin \left(i\right)=i\sinh \left(1\right)$

 $\begin{array}{c}{(-1)}^{i\sinh \left(1\right)}=e^{\left(\ln \left({(-1)}^{i\sin \left(1\right)}\right)\right)}\\ =e^{\left(i\sinh \left(1\right)\ln (-1)\right)}\\ =e^{\left[i\sinh \left(1\right)\left(\ln \left(1\right)+i(\pi \pm 2n\pi )\right)\right]}\\ {(-1)}^{i\sinh \left(1\right)}=e^{\left[-\sinh \left(1\right)(\pi \pm 2n\pi )\right]}\end{array}$

             

 

Hence, 

The value of ${(-1)}^{\sin i}$is$e^{\left[-\sinh \left(1\right)(\pi \pm 2n\pi )\right]}$ Where$n=0,1,2,3,\mathrm{...}$