Q6P - Mathematical Methods in Physical Sciences

Question

Question: Find each of the following in the x + iy form and compare a computer solution.

${\mathbf{tanh}}^{\mathbf{-}\mathbf{1}}\mathbf{(}\mathbf{-}\mathit{i}\mathbf{)}$

Step-by-Step Solution

Answer

The x + iy form is $z=i\left(-\frac{\pi }{4}\pm n\pi \right) n=0,1,2,3,..$

Step 1: Given Information

Given expression is ${\tanh }^{-1}(-i)$.

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

Write the exponential form of the ${\tanh }^{-1}(-i)$.

$\frac{e^{\left(z\right)}-e^{\left(z\right)}}{e^{\left(z\right)}+e^{\left(z\right)}}=-i$

 

Multiply numerator and denominator by.

$$\begin{gathered} e^{\left(2z\right)}-1=-i{e}^{\left(2z\right)}-i \\ {e}^{\left(2z\right)}\left[i+i\right]=\left[1-i\right] \\ {e}^{\left(2z\right)}=\frac{1-i}{1+i} \end{gathered}$$

                                                                                                             …(1)

Take both side in equation (1).

$$\begin{gathered} 2z=\mathrm{In}\left(\frac{1-i}{1+i}\right) \\ 2z=\mathrm{In}\left(\frac{\sqrt{2}{e}^{\left(-\pi i/4\right)}}{\sqrt{2}{e}^{\left(\pi i/4\right)}}\right) \\ 2z=\mathrm{In} e^{\left(-\pi /2\right)} \\ 2z=\mathrm{In}\left(1\right)+i\left(-\frac{\pi }{2}\pm 2n\pi \right) n=0,1,2,3,.. \\ 2z=i\left(-\frac{\pi }{2}\pm 2n\pi \right) n=0,1,2,3,.. \\ 2z=i\left(-\frac{\pi }{4}\pm n\pi \right) n=0,1,2,3,.. \end{gathered}$$

The x + iy form is $z=i\left(-\frac{\pi }{4}\pm n\pi \right) n=0,1,2,3,..$