Q18P - Mathematical Methods in Physical Sciences

Question

Verify each of the following by using equations (11.4), (12.2), and (12.3).

 $\mathbf{tanz}\mathbf{=}\mathbf{tan}\mathbf{(}\mathbf{ }\mathbf{x}\mathbf{+}\mathbf{iy}\mathbf{)}\mathbf{=}\frac{\mathbf{tanx}\mathbf{+}\mathbf{i}\mathbf{ }\mathbf{tanhy}}{\mathbf{1}\mathbf{-}\mathbf{i}\mathbf{ }\mathbf{tanx}\mathbf{ }\mathbf{ }\mathbf{tanhy}}$

Step-by-Step Solution

Answer

The equation $\tan z=\tan ( x+iy)=\frac{\tan x+i \tan hy}{1-i \tan x \tan hy }$is verified using the equations (11.4), (12.2) and (12.3).

Step 1: Given Information

Given equation is,$t\mathrm{an}z=\tan ( x+iy)=\frac{\tan x+i \tan hy}{ 1-i \tan x \tan hy }$ .

Step 2: Definition of Hyperbolic Function.

A hyperbolic function is a representation of the relationship between a point's distances from the origin to the coordinate axes as a function of an angle.

Step 3: Prove the Right Hand Side(RHS) of given equation.

Given the equation is,$\tan z=\tan ( x+iy)=\frac{\tan x+i \tan hy}{1-i \tan x \tan hy }$ .

Write,$\tan z=\tan ( x+yi)$ .

Write,$\tan ( x+yi)=\frac{\sin ( x+yi)}{\cos ( x+yi)}$ .                                               ….(1)

 

Use the following results.

$$\begin{gathered} \sin ( x+yi)=\sin ( x)\cos h( y)+i \cos ( x)\sin h( y) \\ \cos ( x+yi)=\cos ( x)\cos h( y)-i \sin ( x)\sin h( y) \end{gathered}$$

 

Substitute the results in equation (1).

$$\begin{gathered} \tan ( \mathrm{x}+\mathrm{yi})=\frac{\sin ( \mathrm{x})\cosh ( \mathrm{y})+\mathrm{i} \cos ( \mathrm{x})\sinh ( \mathrm{y}) }{\cos ( \mathrm{x})\cosh ( \mathrm{y})-\mathrm{i} \sin ( \mathrm{x})\sinh ( \mathrm{y})}\times \frac{1/\cosh \left(\mathrm{y}\right)}{1/\cosh \left(\mathrm{y}\right)} \\ =\frac{\sin ( \mathrm{x})+\mathrm{i} \cos ( \mathrm{x})\tanh \left(\mathrm{y}\right)}{\cos ( \mathrm{x})\mathrm{i} \sin ( \mathrm{x})\tanh \left(\mathrm{y}\right)}\times \frac{1/\cos \left(\mathrm{x}\right)}{1/\cos \left(\mathrm{x}\right)} \\ =\frac{\tan ( \mathrm{x})+\mathrm{itanh}\left(\mathrm{y}\right)}{1-\mathrm{i} \tan \left(\mathrm{x}\right)\tanh \left(\mathrm{y}\right)} \end{gathered}$$

 

Therefore, Left Hand Side(LHS) is equal to Right Hand Side(RHS).

Hence, the equation is verified.