Q2P - Mathematical Methods in Physical Sciences

Question

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

arctan2i

Step-by-Step Solution

Answer

The values of arctan2i are $$\begin{gathered} -i\ln \left(\frac{1}{\sqrt{3}}\right)+\left(\frac{\pi }{2}\pm 2n\pi \right)\mathrm{and} -i\ln \left(\frac{1}{\sqrt{3}}\right)+\left(\frac{3\pi }{2}\pm 2n\pi \right)\mathrm{where} \\ \mathrm{n}=0,1,2,3,... . \end{gathered}$$.

Step 1: Given information

The given expression is 2i.

Step 2: Definition of Complex Number.

The numbers that are presented in the form of a+ib where, a,b are real numbers and 'i' is an imaginary number called complex numbers.

Step 3: Find the value of 2i.

Consider z=arctan 2i be a complex function.

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

 

Simplify the equation (1).

$$\begin{gathered} e^{\left(zi\right)}-e^{\left(-zi\right)}=-2e^{\left(zi\right)}-e^{\left(-zi\right)} \\ 3e^{\left(2zi\right)}+1=0 ........\left(2\right) \end{gathered}$$

 

Let $u=e^{zi}$and put in equation (2).

$$\begin{gathered} u^2=-\frac{1}{3} \\ u=\pm \frac{i}{\sqrt{3}} \end{gathered}$$

 

Use $u=e^{zi}$ and take natural log both sides.

 

Find the first solution.

$$\begin{gathered} zi=\ln \left(u_1\right) \\ =\mathrm{In}\left(\frac{i}{\sqrt{3}}\right) \\ =\mathrm{In}\left(\frac{\mathrm{i}}{\sqrt{3}}\right)+i\left(\frac{\pi }{2}\pm 2n\pi \right) \\ {z}_1=-i\mathrm{In}\left(\frac{\mathrm{i}}{\sqrt{3}}\right)+\left(\frac{\mathrm{\pi }}{2}\pm 2\mathrm{n\pi }\right) \end{gathered}$$

Find the second solution.

$$\begin{gathered} zi=\mathrm{In}\left({\mathrm{u}}_2\right) \\ =\mathrm{In}\left(\frac{-\mathrm{i}}{\sqrt{3}}\right) \\ =\mathrm{In}\left(\frac{-\mathrm{i}}{\sqrt{3}}\right)+\mathrm{i}\left(\frac{3\mathrm{\pi }}{2}\pm 2\mathrm{n\pi }\right) \\ {\mathrm{z}}_2=-i\mathrm{In}\left(\frac{-\mathrm{i}}{\sqrt{3}}\right)+\left(\frac{3\mathrm{\pi }}{2}\pm 2\mathrm{n\pi }\right) \\ \mathrm{Hence} \mathrm{the} \mathrm{values} \mathrm{of} \arctan 2\mathrm{i} \mathrm{are} -i\mathrm{In}\left(\frac{-\mathrm{i}}{\sqrt{3}}\right)+\left(\frac{3\mathrm{\pi }}{2}\pm 2\mathrm{n\pi }\right) \mathrm{and} -\mathrm{iIn}\left(\frac{-\mathrm{i}}{\sqrt{3}}\right)+\left(\frac{3\mathrm{\pi }}{2}\pm 2\mathrm{n\pi }\right) \\ \mathrm{where} \mathrm{n}=0,1,2,3,... . \end{gathered}$$

Hence the values of are and where