Q20P - Mathematical Methods in Physical Sciences

Question

Verify the formulas in Problems 17 to 24.

$\mathit{s}\mathit{i}\mathit{n}{\mathit{h}}^{\mathbf{-}\mathbf{1}}\mathit{z}\mathbf{=}\mathit{l}\mathit{n}\mathbf{(}\mathit{z}\mathbf{\pm }\sqrt{{\mathbf{z}}^{\mathbf{2}}\mathbf{+}\mathbf{1}}\mathbf{)}$

Step-by-Step Solution

Answer

The formula${\sinh }^{-1}\mathrm{z}=\ln (\mathrm{z}\pm \sqrt{{\mathrm{z}}^2+1})$ is verified.

Step 1: Given Information

Given formula is ${\sinh }^{-1}\mathrm{z}=\ln (\mathrm{z}\pm \sqrt{{\mathrm{z}}^2+1})$ .

Step 2: Definition of Trigonometric equation.

A trigonometric equation is one that has one or more trigonometric ratios with unknown angles.

Step 3: Use exponential form to expand the equation

Given the function ${\sinh }^{-1}\mathrm{z}=\ln (\mathrm{z}\pm \sqrt{{\mathrm{z}}^2+1})$ .

Write the exponential form of the $z=\sin h( w).$

 $$\begin{gathered} z\frac{e^{\left(w\right)}-e^{\left(-w\right)}}{2} \\ {e}^{\left(2w\right)}-2e^{\left(w\right)}-1=0 \end{gathered}$$                                                                                                 …(1)

 

Let $u=e^{\left(w\right)}$ in equation (2).

  $u^2-2zu-1=0$                                                                                                      …(2)

 

The coefficient of equation is as follows.

$$\begin{gathered} a=1 \\ b=-2z \\ c=-1 \end{gathered}$$

 

Use quadratic formula to find roots of equation (2).

  $$\begin{gathered} u=\frac{-b\pm \sqrt{b^2-4aac}}{2a} \\ =\frac{2z\sqrt{{\left(-2z\right)}^2+4}}{2} \\ =z\pm \sqrt{z^2+1} \end{gathered}$$

Replace value of u by $u=e^{\left(w\right)}$ .

$w =\ln \left(z\pm \sqrt{z^2+1}\right)$

Replace value of w by $w=sin{h}^{-1}\left(z\right).$ .

$\left(z\right)=i\ln \left(z\pm \sqrt{z^2+1}\right)$

Hence the formula is verified.