Proof Help: Solving Griffith's Problem 9.15 for Ae^iax + Be^ibx = Ce^icx

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In summary, the conversation discusses proving the equality of constants a, b, and c and the sum of amplitudes A and B with C in the equation Ae^iax + Be^ibx = Ce^icx. A hint is given to set x=0 to prove A+B=C, but it is noted that this does not necessarily prove a=b=c. To prove this, the independence of e^ix and e^x for x\in \mathbb {R} is used. Finally, it is stated that Ae^{ix}+Be^{x}=0\Leftrightarrow A=B=0.
  • #1
pt176900
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from Griffith's, problem 9.15: 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

I'm definitely confused on where to begin my manipulation. It seems quite reasonable to meet that the constants should be equal, and the amplitudes should sum up to C but I don't know how to get there mathematically.

Can someone give me a hint to get me started?
 
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  • #3
if I make x = 0 then the relation a=b=c need not hold.

By this I mean that e^n*0 = 1 for all n, hence a, b, and c can be any real number.

So basically I'd trade proving a=b=c for A+B=C
 
  • #4
ok so wait... I can use x = 0 to prove A+B=C (I did it by contradiction ie assume A+B != C for all x. then insert x=0 and you get A+B=C - a contradiction).

then can I use the fact that A+B = C to prove that a=b=c?
 
  • #5
Nope.That x=0 will probe that A+B=C.Now u'll have to make use of the independence of the [itex] e^{ix}[/itex] and [itex] e^{x} [/itex] for [itex] x\in \mathbb {R} [/itex].

Daniel.
 
  • #6
I'm sorry but I don't understand what you mean by the independence of e^ix and e^x.
 
  • #7
[tex] Ae^{ix}+Be^{x}=0\Leftrightarrow A=B=0 [/tex]


Daniel.
 

Related to Proof Help: Solving Griffith's Problem 9.15 for Ae^iax + Be^ibx = Ce^icx

1. What is a proof in science?

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5. Why is proof important in science?

Proof is important in science because it allows us to validate and support our ideas and theories. It helps us to distinguish between fact and fiction, and to understand the natural world in a more accurate and objective way. Proof also allows for the advancement of scientific knowledge and the development of new technologies and innovations.

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