Chapter 2: Q17P (page 76)

Show that tan z never takes the values $\pm i$ . Hint: Try to solve the equation tan z= iand find that it leads to a contradiction.

Short Answer

Expert verified

It has been proved thattan z never takes the values $\pm i$ .

Step by step solution

Given Information.

The given expression is tan z.

Meaning of rectangular form.

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

Solve the equation tan z = i.

Solve the given equation tan z = i in the hint.

$\frac{[e^{(z i)} - e^{(- z i)}]}{i [e^{(z i)} + e^{(- z i)}]} = i e^{(z i)} - e^{(- z i)} = - e^{(z i)} - e^{(- z i)} 2 e^{(z i)} = 0$

Previous equation has no exact solution because if we try to take it gives us infinity which means it's not valid for tan (z) to take $\pm i$ .

$\frac{[e^{(z i)} - e^{(- z i)}]}{i [e^{(z i)} + e^{(- z i)}]} = i e^{(z i)} - e^{(- z i)} = - e^{(z i)} - e^{(- z i)} 2 e^{(- 2 z)} = 0$

Previous equation has no exact solution because if we try to take it gives us infinity which means it's not valid for tan (z) to take $\pm i$ .

The final equation is $e^{(2 z)} = 0$ which has no exact value.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Show that tan z never takes the values $\pm i$ . Hint: Try to solve the equation tan z= iand find that it leads to a contradiction.

Short Answer

Step by step solution

Given Information.

Meaning of rectangular form.

Solve the equation tan z = i.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Physical Quantities And Units

Kinematics Physics

Classical Mechanics

Rotational Dynamics

Geometrical and Physical Optics

Electric Charge Field and Potential

Study anywhere. Anytime. Across all devices.

Show that tan z never takes the values ±i. Hint: Try to solve the equation tan z= iand find that it leads to a contradiction.

Short Answer

Step by step solution

Given Information.

Meaning of rectangular form.

Solve the equation tan z = i.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Physical Quantities And Units

Kinematics Physics

Classical Mechanics

Rotational Dynamics

Geometrical and Physical Optics

Electric Charge Field and Potential

Study anywhere. Anytime. Across all devices.

Show that tan z never takes the values $\pm i$ . Hint: Try to solve the equation tan z= iand find that it leads to a contradiction.