Chapter 2: Q2P (page 76)

Find each of the following in the x+ iyform and compare a computer solution.
arctan2i

Short Answer

Expert verified

The values of arctan2i are $- i \ln (\frac{1}{\sqrt{3}}) + (\frac{π}{2} \pm 2 n π) and - i \ln (\frac{1}{\sqrt{3}}) + (\frac{3 π}{2} \pm 2 n π) where n = 0, 1, 2, 3, . . . .$ .

Step by step solution

Given information

The given expression is 2i.

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.

Find the value of 2i.

Consider z=arctan 2i be a complex function.

$\tan (z) = 2 i - i (\frac{e^{(z i)} - e^{(- z i)}}{e^{(z i)} + e^{(- z i)}}) = 2 i . . . . . . . . (1)$

Simplify the equation (1).

$e^{(z i)} - e^{(- z i)} = - 2 e^{(z i)} - e^{(- z i)} 3 e^{(2 z i)} + 1 = 0 . . . . . . . . (2)$

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

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

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

Find the first solution.

$z i = \ln (u_{1}) = In (\frac{i}{\sqrt{3}}) = In (\frac{i}{\sqrt{3}}) + i (\frac{π}{2} \pm 2 n π) z_{1} = - i In (\frac{i}{\sqrt{3}}) + (\frac{π}{2} \pm 2 nπ)$

Find the second solution.

$z i = In (u_{2}) = In (\frac{- i}{\sqrt{3}}) = In (\frac{- i}{\sqrt{3}}) + i (\frac{3 π}{2} \pm 2 nπ) z_{2} = - i In (\frac{- i}{\sqrt{3}}) + (\frac{3 π}{2} \pm 2 nπ) Hence the values of \arctan 2 i are - i In (\frac{- i}{\sqrt{3}}) + (\frac{3 π}{2} \pm 2 nπ) and - iIn (\frac{- i}{\sqrt{3}}) + (\frac{3 π}{2} \pm 2 nπ) where n = 0, 1, 2, 3, . . . .$

Hence the values of are and where

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Find each of the following in the x+ iyform and compare a computer solution.
arctan2i

Short Answer

Step by step solution

Given information

Definition of Complex Number.

Find the value of 2i.

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Find each of the following in the x+ iyform and compare a computer solution.arctan2i

Short Answer

Step by step solution

Given information

Definition of Complex Number.

Find the value of 2i.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Atoms and Radioactivity

Geometrical and Physical Optics

Quantum Physics

Electricity

Turning Points in Physics

Waves Physics

Study anywhere. Anytime. Across all devices.

Find each of the following in the x+ iyform and compare a computer solution.
arctan2i