Q9P - Mathematical Methods in Physical Sciences

Question

Find each of the following in the $\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$ form and compare a computer solution.

 $\mathit{t}\mathit{a}\mathit{n}{\mathit{h}}^{\mathbf{-}}\left(i\sqrt{3}\right)$.

Step-by-Step Solution

Answer

The $x+iy$ form of the given equation  $\tan h^{-1}\left(i\sqrt{3}\right)$ is $i\left(\frac{\pi }{3}\pm n\pi \right)$ 

Step 1: Given Information.

The given expression is, $\tan h^{-1}\left(i\sqrt{3}\right)$ .

Step 2: Meaning of rectangular form.

Represent the complex number in rectangular form means writing the given complex number in the form of x+iy in which x is the real part and y is the imaginary part.

Step 3: Convert into polar form.

Consider the complex number is, $z=\tan h^{-1}\left(i\sqrt[ ]{3}\right)$ .

Rewrite the above expression.

 $\tan h z=\left(i\sqrt{3}\right)$

Write the formula for  $\left(\theta \right)$.

$\frac{e^{\left(z\right)}-e^{\left(-z\right)}}{e^{\left(z\right)}+e^{\left(-z\right)}}=i\sqrt{3}$

  

Multiply it by $\frac{e^{\left(z\right)}}{e^{\left(z\right)}}$.

$$\begin{gathered} e^{\left(2z\right)}-1=i{\sqrt{3e}}^{2z}+i\sqrt{3} \\ {e}^{\left(2z\right)}\left[1-i\sqrt{3}\right]=\left[1+i\sqrt{3}\right] \\ e^{\left(2z\right)}=\frac{1+i\sqrt{3}}{1i\sqrt{3}} \end{gathered}$$

 

Step 4: Convert in rectangular form

Take the logarithm function both sides.

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

 

$2z=i\left(\frac{2\pi }{3}\pm 2n\pi \right)$                Where  $n=0,1,2,3,..$

$z=i\left(\frac{\pi }{3}\pm n\pi \right)$ .                    Where $n=0,1,2,3,..$$n=0,1,2,3,...$ 

 

Hence the general solution is $z=i\left(\frac{\pi }{3}\pm n\pi \right)$ .