Identity for exponential function with imaginary arguments

In summary, the identity Exp[iw/2]-Exp[iw/2]=Exp[iw]-1 can be proven by simply multiplying the left side by Exp[iw/2] and using the exponent rule. This simplifies the left side to 2isin(w/2), which is equal to the right side, cos(w)+isin(w)-1.
  • #1
ParityCheck
2
0
I have seen the following identity used.
Exp[iw/2]-Exp[iw/2]=Exp[iw]-1
I can't find this in any book and I can't prove it myself.
The left side equals 2isin(w/2)
The right side equals cos(w)+isin(w)-1
On the face of it, that seems to make the identity absurd
How can one go about proving such an identity? Or is it wrong?
 
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  • #2
Welcome to PF!

ParityCheck said:
Exp[iw/2]-Exp[iw/2]=Exp[iw]-1
…
How can one go about proving such an identity? Or is it wrong?

Hi ParityCheck ! Welcome to PF! :smile:

(eiw/2 - e-iw/2)eiw/2 = eiw - 1 :wink:
 
  • #3
Thanks. I was making it way too complicated with trig identities and series and...
 

FAQ: Identity for exponential function with imaginary arguments

1. What is an exponential function with imaginary arguments?

An exponential function with imaginary arguments is a mathematical function of the form f(x) = e^(ix) where i is the imaginary unit (√-1) and e is Euler's number (2.71828...). This type of function is used to describe exponential growth or decay in situations where the input variable x is a complex number.

2. How do you graph an exponential function with imaginary arguments?

To graph an exponential function with imaginary arguments, you can use the properties of exponents to simplify the expression. For example, e^(ix) can be rewritten as cos(x) + i*sin(x), which can then be graphed as a sinusoidal curve in the complex plane.

3. What is the relationship between exponential functions with imaginary arguments and trigonometric functions?

Exponential functions with imaginary arguments are closely related to trigonometric functions. In fact, they can be expressed in terms of trigonometric functions using Euler's formula: e^(ix) = cos(x) + i*sin(x). This relationship is useful in solving complex equations and studying periodic phenomena.

4. How is the identity for exponential function with imaginary arguments derived?

The identity for exponential function with imaginary arguments is derived from Euler's formula and the properties of exponents. By expanding e^(ix) using Maclaurin series and simplifying, we can obtain the identity: e^(ix) = cos(x) + i*sin(x).

5. What are some real-world applications of exponential functions with imaginary arguments?

Exponential functions with imaginary arguments are used in a variety of fields, including physics, engineering, and finance. They can be used to model phenomena such as electromagnetic waves, quantum mechanics, and compound interest. They are also used in signal processing and electrical circuit analysis.

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