Q15P - Mathematical Methods in Physical Sciences

Question

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

siniz = i sin z

Step-by-Step Solution

Answer

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

Step 1: Given Information

Given equation is sinh iz = i sin 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 sinh iz = i sin z.

The exponential form of the given equation is,

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

 ….(1)

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

 

Solve sin (iz) to prove the given equation.

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

Multiply numerator and denominator by i.

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

Hence, the equation is verified.