Q17P - Mathematical Methods in Physical Sciences

Question

Verify the formulas in Problems 17 to 24.

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

Step-by-Step Solution

Answer

The formula is verified $arcsin z=-i ln(\mathrm{iz}\pm \sqrt{1-{\mathrm{z}}^2})$

Step 1: Given Information

Given formula is $arcsin z=-i ln(\mathrm{iz}\pm \sqrt{1-{\mathrm{z}}^2})$

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.

$wi=ln(zi\pm \sqrt{1-z^2})$Given the function is, $arcsin z=-i ln(\mathrm{iz}\pm \sqrt{1-{\mathrm{z}}^2})$ .

Write the exponential form of the .

$$\begin{gathered} z=\sin \left(w\right) \\ =\frac{e^{\left(wi\right)}-e^{\left(-wi\right)}}{2i} ......\left(1\right) \end{gathered}$$

Rewrite the equation (1).

$$\begin{gathered} e^{\left(wi\right)}-e^{\left(-wi\right)}=2zi \\ {e}^{\left(2wi\right)}-2zi{e}^{\left(wi\right)}-1=0 .....\left(2\right) \end{gathered}$$

 

Let in equation (2).

$u^2-2ziu-1=0 ......\left(3\right)$

 

Use quadratic formula to find roots of equation (3).

The coefficient of equation is as follows.

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

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

$wi=ln(zi\pm \sqrt{1-z^2})$

 

Replace value of w by $w=arc( \sin z )$.

$arcsin z=-i ln(\mathrm{iz}\pm \sqrt{1-{\mathrm{z}}^2})$

Hence the formula is verified.