Q19P - Mathematical Methods in Physical Sciences

Question

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

$\mathit{t}\mathit{a}\mathit{n}\mathit{h}\mathbf{ }\mathit{z}\mathbf{=}\frac{\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{h}\mathbf{ }\mathbf{x}\mathbf{+}\mathbf{i}\mathbf{ }\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{ }\mathbf{y}}{\mathbf{1}\mathbf{+}\mathbf{i}\mathbf{ }\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{h}\mathbf{ }\mathbf{x}\mathbf{ }\mathbf{ }\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{ }\mathbf{y}\mathbf{ }}$

Step-by-Step Solution

Answer

The equation $tanh z=\frac{\tanh \mathrm{x}+\mathrm{i} \tan \mathrm{y}}{1+\mathrm{i} \tanh \mathrm{x} \tan \mathrm{y} }$ is verified using the equations (11.4), (12.2) and (12.3).

Step 1: Given information

The given equation is,$tanh z=\frac{\tanh \mathrm{x}+\mathrm{i} \tan \mathrm{y}}{1+\mathrm{i} \tanh \mathrm{x} \tan \mathrm{y} }$ .

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.

Step 3: Use the standard form of complex number to expand the equation

The given function is,

$tanh z=\frac{\tanh \mathrm{x}+\mathrm{i} \tan \mathrm{y}}{1+\mathrm{i} \tanh \mathrm{x} \tan \mathrm{y} }$                                                                                            …. (1)

 

Put $z=x+iy$ in equation (1).

 $$\begin{gathered} \tan z=\tan \left(x-yi\right) \\ \tan h\left(x+iy\right)=\frac{\sin h\left(x+yi\right)}{\cos h\left(x+yi\right)} \\ =\frac{\sin \left(x\right)\cos \left(y\right)+i \cos h\left(x\right)\sin \left(y\right)}{\cos h\left(x\right)\cos \left(y\right)+i \sin h\left(x\right)\sin \left(y\right)} \end{gathered}$$

 

Divide numerator and denominator by $\left[\cos h\left(x\right).\cos \left(y\right)\right]$ .

$$\begin{gathered} \tan h\left(x+iy\right)=\frac{\sin \left(x\right)\cos \left(y\right)+i\cos h\left(x\right)\sin \left(y\right)}{\cos h\left(x\right)\cos \left(y\right)+i\sin h\left(x\right)\sin \left(y\right)} \\ =\frac{\left[\sin h\left(x\right)\cos \left(y\right)+i\cos h\left(x\right)\sin \left(y\right)\right]/\left[\cos h\left(x\right)\cos \left(y\right)\right]}{\left[\cos h\left(x\right)\cos \left(y\right)+i \sin h\left(x\right)\sin \left(y\right)\right]/\left[\cos h\left(x\right)\cos \left(y\right)\right]} \end{gathered}$$

$$\begin{gathered} \tan h\left(x+iy\right)=\frac{\left({\displaystyle \frac{\sin h\left(x\right)\cos \left(y\right)}{\cos \left(x\right)\cos \left(y\right)}}\right)+\left({\displaystyle \frac{i \cos h\left(x\right)\sin \left(y\right)}{\cos h\left(x\right)\cos \left(y\right)}}\right)}{\left({\displaystyle \frac{\cos h\left(x\right)\cos \left(y\right)}{\cos h\left(x\right)\cos \left(y\right)}}\right)+\left({\displaystyle \frac{i \sin h\left(x\right)\sin \left(y\right)}{\cos h\left(x\right)\cos \left(y\right)}}\right)} \\ =\frac{\tan h\left(x\right)+i \tan \left(y\right)}{1+i \tan h\left(x\right)\tan \left(y\right)} \end{gathered}$$

Hence the equation is verified.