Proving the Addition Formula for Complex Functions

In summary, the given equation can be simplified by dividing both sides by the first half of the righthand side and using the identity eiC + e-iC = 2cosC. This simplification is not explicitly stated in the book, but can be easily derived using the given hint.
  • #1
kasse
384
1
How can I show that

[tex]Aexpi(\omega t - kx + \phi_{1}) + Aexpi(\omega t + kx + \phi_{2}) = 2Aexpi(\omega t + \frac{\phi_{1} + \phi_{2}}{2})cos(kx - \frac{\phi_{1} - \phi_{2}}{2})[/tex]? My book states this without proof, as if it was obvious.
 
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  • #2
Hi kasse! :smile:

Hint: eiA + B = eiA + eiB, so divide by the first half of the righthand side, and use eiC + e-iC = 2cosC. :wink:
 
  • #3
Danke schön
 
  • #4
tiny-tim said:
Hi kasse! :smile:

Hint: eiA + B = eiA + eiB, so divide by the first half of the righthand side, and use eiC + e-iC = 2cosC. :wink:
Oh, dear, oh, dear, oh, dear, tiny-tim! eiA + B = (eiA)(eiB).
 
  • #5
oops!

HallsofIvy said:
Oh, dear, oh, dear, oh, dear, tiny-tim! eiA + B = (eiA)(eiB).

Geman for oops! :redface:
 
  • #6
eiA + B = (eiA)(eB)...?
 

1. What are complex functions?

Complex functions are mathematical functions that involve complex numbers, which are numbers that have both a real and imaginary component. They are commonly used in mathematics and engineering to model real-world phenomena.

2. How do you add two complex functions?

To add two complex functions, you simply add the real parts together and add the imaginary parts together. For example, if you have two complex functions f(x) = 2 + 3i and g(x) = 4 + 2i, their sum would be h(x) = (2+4) + (3+2)i = 6 + 5i.

3. Can you add two complex functions with different variables?

No, in order to add two complex functions, they must have the same variable. This is because the real and imaginary parts of complex numbers are dependent on the same variable.

4. What is the result of adding two complex functions with the same variable?

The result of adding two complex functions with the same variable is another complex function with the same variable. This means that the sum of two complex functions is also a complex function.

5. Are there any special rules for adding two complex functions?

Yes, there are some special rules for adding two complex functions, such as the commutative and associative properties. These properties state that the order in which you add the functions does not affect the result, and that you can group the functions in different ways and still get the same result. Additionally, the additive inverse property holds, meaning that the sum of a complex function and its additive inverse is equal to 0.

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