Seemingly simple proof I can't seem to get

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Homework Help Overview

The discussion revolves around a mathematical proof involving complex exponentials, specifically the equation Ae^(iax) + Be^(ibx) = Ce^(icx). Participants are tasked with demonstrating that a = b = c and A + B = C for nonzero constants A, B, C, a, b, and c, valid for all x.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants explore various methods, including evaluating the equation at specific values of x and questioning the implications of the equation being true for all x. Some suggest using the Euler Formula, while others express uncertainty about its relevance.

Discussion Status

The discussion is ongoing, with participants raising questions about the assumptions underlying the problem and exploring different approaches. There is no explicit consensus on the best method to proceed, but various lines of reasoning are being examined.

Contextual Notes

Participants note potential restrictions on the constants involved, suggesting that without additional constraints, the relationships could lead to trivial solutions. The original question emphasizes the requirement for the equation to hold for all x.

mathlete
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I have:

Ae^(iax)+Be^(ibx)=Ce^(icx)

I have to show a=b=c and A+B=C but I can't... I've tried some standard tricks like squaring both sides, taking the derivative, then playing around with the equations but I can't get anything to stick. Any ideas?
 
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It has to be true for all x, right? What about your favorite value of x?
 
Hurkyl said:
It has to be true for all x, right? What about your favorite value of x?
Yes, sorry, for all x... and I'm afraid I don't understand what you mean by my favorite value of x (x=0?)
 
There surely have to be some more restrictions than this, otherwise A+B=0=C, a=b=anything, and c is anything else not equal to a
 
matt grime said:
There surely have to be some more restrictions than this, otherwise A+B=0=C, a=b=anything, and c is anything else not equal to a
Question word for word...

"Suppose Ae^(iax)+Be^(ibx)=Ce^(icx), for some nonzero constants A, B, C, a, b, c, and for all x. Prove that a=b=c and A+B=C"

Any ideas?
 
The Euler Formula expansion is key.
 
No it's not. You just have to use the fact it's true for all x.

I'm afraid I don't understand what you mean by my favorite value of x (x=0?)
The equation is true for x=0, right? What does the equation look like when x=0?
 
Last edited:
Hurkyl said:
No it's not. You just have to use the fact it's true for all x.


The equation is true for x=0, right? What does the equation look like when x=0?
Nevermind then... I was thinking if it had to be true for any given constant value of x that would be the way to go about it.
 
Actually, what do you mean by Euler's expansion? I was thinking e^(ix) = cos x + i sin x... though I don't remember if I've ever heard it named that.

If so, then this does give you a shortcut if you know the relevant theorems... but the same shortcut works directly for the problem at hand!
 

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