Q16P - Mathematical Methods in Physical Sciences

Question

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

$\mathbf{ }\mathbf{tani}\mathbf{ }\mathbf{z}\mathbf{=}\mathbf{i}\mathbf{ }\mathbf{tanhz}$

Step-by-Step Solution

Answer

The equation $\mathrm{tani} \mathrm{z}=\mathrm{i} \mathrm{tanhz}$ is verified using the equations (11.4), (12.2) and (12.3).

Step 1: Given Information

Given equation is$\mathrm{tani} \mathrm{z}=\mathrm{i} \mathrm{tanhz}$.

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: Simplify and Prove the Right Hand Side (RHS) of given equation.

Given the equation is,$\mathrm{tani} \mathrm{z}=\mathrm{i} \mathrm{tanhz}$ .

Write $\tan \mathrm{z}$as $\tan ( z)=\frac{\sin ( z)}{\cos ( z) }$

Let $z\to zi$ than$\tan ( zi)=\frac{\sin ( zi) }{\cos ( zi) } \dots .\left(1\right)$                    Use the following results.

$$\begin{gathered} \cos ( iz)=\cos h( z) \\ \sin ( iz)=i \sin h( z) \end{gathered}$$

 

Substitute the results in equation (1).

$$\begin{gathered} \tan ( iz)=\frac{i \sin h( z)}{\cos h( z)} \\ \tan ( iz)=i \tan h( z) \end{gathered}$$

 

Hence, the equation is verified.