Q3P - Mathematical Methods in Physical Sciences

Question

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

 $\mathbf{sinh}\mathbf{ }\mathbf{z}\mathbf{=}\mathbf{sinh}\mathbf{ }\mathbf{x}\mathbf{ }\mathbf{cos}\mathbf{ }\mathbf{y}\mathbf{+}\mathbf{i}\mathbf{ }\mathbf{cosh}\mathbf{ }\mathbf{x}\mathbf{ }\mathbf{sin}\mathbf{ }\mathbf{y}$

Step-by-Step Solution

Answer

The equation sin h z=sinh x cos y+i cosh x sin y is verified using the equations (11.4), (12.2) and (12.3). 

Step 1: Given information

Thegiven the function sin h z= sin h x cos y +i cosh x sin y.

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.

Relation between the exponential and polar form is$\mathit{r}{\mathit{e}}^{\mathbf{i}\mathbf{\theta }}\mathbf{ }\mathbf{=}\mathit{r}\mathbf{ }\mathit{c}\mathit{o}\mathit{s}\mathbf{ }\mathit{\theta }\mathbf{+}\mathit{i}\mathit{r}\mathbf{ }\mathit{s}\mathit{i}\mathit{n}\mathbf{ }\mathit{\theta }$.

Step 3: Use exponential and polar form to expand the equation

The exponential form of the given equation is,

$\sin h z=\frac{e^z-e^{-z}}{2}$                                                                                                         …. (1)

 

Let z=x+yi and put in equation (1).

$\sin h z=\frac{e^{\left(x+yi\right)}-e^{\left(-x+yi\right)}}{2}$

 

Write x+yi as i(-xi+y).

$$\begin{gathered} \sin h z=\frac{e^{\left[\left(-xi+y\right)i\right]}-e^{\left[\left(xi+y\right)i\right]}}{2} \\ \sin h z=\frac{e^{\left(-xi\right)i}.e^{\left(yi\right)}-e^{\left(xi\right)i}.e^{\left(-yi\right)}}{2} \end{gathered}$$

 

Convert exponential form into polar form.

$\sin h z=\frac{\left[\cos xi-\sin ix\right] \left[\cos y+i \sin y\right]-\left[\cos xi+\sin ix\right] \left[\cos y-i \sin y\right]}{2}$                    …. (2) 

Step 4: Replace cos xi by cosh x and sin xi by i sinh x

 Replace cos xi by cosh xand sin xi by i sinh x in equation (2).

$$\begin{gathered} \sinh \mathrm{z}=\frac{\left[\cosh \mathrm{x}+\sinh \mathrm{x}\right] \left[\cos \mathrm{y}+\mathrm{i} \sin \mathrm{y}\right]-\left[\cosh \mathrm{x}-\sinh \mathrm{x}\right] \left[\cos \mathrm{y}-\mathrm{i} \sin \mathrm{y}\right]}{2} \\ =\left\{\begin{array}{c}\frac{\left[\cosh \mathrm{x}.\cos \mathrm{y}+\mathrm{i} \sinh \mathrm{x}. \sin \mathrm{y}+\mathrm{i} \cosh \mathrm{x}. \sin \mathrm{y}+\sin \mathrm{x}. \cos \mathrm{y}\right]}{2}\\ \frac{\left[\cosh \mathrm{x}.\cos \mathrm{y}+\mathrm{i} \sinh \mathrm{x}. \sin \mathrm{y}-\mathrm{i} \cosh \mathrm{x}. \sin \mathrm{y}-\sin \mathrm{x}. \cos \mathrm{y}\right]}{2}\end{array}\right\} \\ \mathrm{Cancel} \mathrm{similar} \mathrm{terms}. \\ \sinh \mathrm{z}=\frac{2 \sinh \mathrm{x} \cos \mathrm{y}}{2}+\frac{2\mathrm{i} \sin \mathrm{y} \cosh \mathrm{x}}{2} \\ \sinh \mathrm{z}=\sinh \mathrm{x} \cos \mathrm{y}+\mathrm{i} \sin \mathrm{y} \cosh \mathrm{x} \\ \mathrm{Hence} \mathrm{the} \mathrm{equation} \mathrm{is} \mathrm{verified}. \end{gathered}$$