Exponential Functions In Complex Analysis

In summary, the conversation discusses the correct answer for the equation e^(5pi/4) = (1-i)/(-sqrt(2)), with a suggestion to check the sign of the imaginary part and a reminder to show the process of arriving at the answer. It also mentions the difficulty of having a real number on one side of the equation and an imaginary number on the other. The possibility of using e^(5pi i/4) and e^(3pi i/4) instead is brought up.
  • #1
calculuskatie
2
0
Can someone please tell me if I have the correct answer for this one?

e^(5pi/4) = (1-i)/(-sqrt(2))

Thanks...
 
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  • #2
You might want to check the sign of the imaginary part...
Drawing a picture might help.
 
  • #3
I guess it should be e^(3pi/4)
 
  • #4
Stop guessing! Show us how you arrived at either of those answer, please.

I will point out that whether you say
[tex]e^{5\pi/4}= \frac{1- i}{-\sqrt{2}}[/itex]
or
[tex]e^{3pi/4}= \frac{1- i}{-\sqrt{2}}[/itex]
You run into the rather sizeable difficulty that the number on the left of the equal sign is real while the number on the right is not!

Is it at all possible that you meant [itex]e^{5\pi i/4}[/itex] and [itex]e^{3\pi i/4}[/itex]?
 

Related to Exponential Functions In Complex Analysis

1. What is an exponential function in complex analysis?

An exponential function in complex analysis is a function of the form f(z) = ez, where e is Euler's number and z is a complex number. It is a special type of function that has many unique properties, including being its own derivative and being periodic in the imaginary direction.

2. How are exponential functions used in complex analysis?

Exponential functions are used in complex analysis to study and analyze complex numbers and their properties. They can be used to solve differential equations, calculate limits, and represent complex numbers in polar form.

3. What is the relationship between exponential functions and trigonometric functions in complex analysis?

There is a close relationship between exponential functions and trigonometric functions in complex analysis. This is because the imaginary unit i can be expressed as eπi/2, and trigonometric functions can be written in terms of complex exponential functions. This connection is known as Euler's formula.

4. Can exponential functions in complex analysis have complex inputs and outputs?

Yes, exponential functions in complex analysis can have both complex inputs and outputs. This is because complex numbers can be raised to complex powers, resulting in another complex number.

5. What is the significance of exponential functions in complex analysis?

Exponential functions play a crucial role in complex analysis as they allow for the representation and manipulation of complex numbers in a simple and elegant way. They also have many applications in physics, engineering, and other fields.

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