Q14P - Mathematical Methods in Physical Sciences

Question

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

 

$\mathbf{sin} \mathit{i}\mathit{z}\mathbf{=}\mathit{i}\mathbf{ }\mathbf{sinh}\mathbf{ }\mathit{z}$

Step-by-Step Solution

Answer

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

Step 1: Given Information

Given equation is $\sin iz=i \sinh z$

Step 2: Definition of Hyperbolic Function.

The term "Hyperbolic Function" refers to the relationship between a point on a hyperbola's distance from its origin and its coordinate axes, expressed as a function of an angle.

Step 3: Use exponential form to expand the equation

Given the equation is $\sin iz=i \sinh z$.

 

The exponential form of the given equation is,

$\sin \left(z\right)=\frac{e^{\left(zi\right)}-e^{\left(-zi\right)}}{2i} ...\left(1\right)$                                                                                         

Let $z\to zi$ than $\sin \left(iz\right)=\frac{e^{\left(zi.i\right)}-e^{\left(-zi.i\right)}}{2i}$.

 

Solve $\sin \left(iz\right)$ to prove the given equation.

$$\begin{gathered} \sin \left(iz\right)=\frac{e^{\left(zi.i\right)}-e^{\left(-zi.i\right)}}{2i} \\ \sin \left(iz\right)=\frac{e^{\left(-z\right)}-e^{\left(z\right)}}{2i} \\ \sin \left(iz\right)=\frac{i}{2}\left[\exp (-z)-\exp \left(z\right)\right] \\ \sin \left(iz\right)=\frac{i}{2}\left[\exp \left(z\right)-\exp (-z)\right] \\ \sin \left(iz\right)=\left(i\right)\frac{\exp \left(z\right)-\exp (-z)}{2} \\ \sin \left(iz\right)=i \sinh \left(z\right) \end{gathered}$$

Hence, the equation is verified.