Q17P - Mathematical Methods in Physical Sciences

Question

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

 $\mathbf{ }\mathit{t}\mathit{a}\mathit{n}\mathit{h}\mathit{i}\mathbf{ }\mathit{z}\mathbf{=}\mathit{i}\mathbf{ }\mathit{t}\mathit{a}\mathit{n}\mathit{z}$

Step-by-Step Solution

Answer

The equation  $\tan hi z=i \tan z$is verified using the equations (11.4), (12.2) and (12.3).

Step 1: Given Information

Given equation is$\tan hi z=i \tan z$

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,$\mathrm{tanhi} \mathrm{z}=\mathrm{i} \mathrm{tanz}$ .

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

Let$z\to zi$ than $\tanh ( \mathrm{zi})=\frac{\sinh ( \mathrm{zi})}{\cosh ( \mathrm{zi})}$                                       ….(1)

Solve Left hand side(LHS) i.e,.

$$\begin{gathered} \tanh ( \mathrm{zi})=\frac{\sinh ( \mathrm{zi})}{\cosh ( \mathrm{zi})} \\ =\left(\frac{e^{\left(zi\right)}-e^{\left(-zi\right)}}{2}\right)\left(\frac{2}{e^{\left(zi\right)}+e^{\left(-zi\right)}}\right) \end{gathered}$$

 

Multiply numerator and denominator by

$$\begin{gathered} \tanh ( \mathrm{zi}) =\left(\frac{{\mathrm{e}}^{\left(\mathrm{zi}\right)}-{\mathrm{e}}^{\left(-\mathrm{zi}\right)}}{2}\right)\left(\frac{2}{{\mathrm{e}}^{\left(\mathrm{zi}\right)}+{\mathrm{e}}^{\left(-\mathrm{zi}\right)}}\right)\times \frac{\mathrm{i}}{\mathrm{i}} \\ \tanh ( \mathrm{zi})=\left(\frac{{\mathrm{e}}^{\left(\mathrm{zi}\right)}-{\mathrm{e}}^{\left(-\mathrm{zi}\right)}}{2\mathrm{i}}\right)\left(\frac{2}{{\mathrm{e}}^{\left(\mathrm{zi}\right)}+{\mathrm{e}}^{\left(-\mathrm{zi}\right)}}\right) \\ \tanh ( \mathrm{zi})=\mathrm{i} \sin \left(\mathrm{z}\right)\cos \left(\mathrm{z}\right) \end{gathered}$$

 

Hence, the equation is verified.