Need some help with a proof

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  • #1
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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 dont know how to get there mathematically.

Can someone give me a hint to get me started?
 

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  • #3
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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
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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
dextercioby
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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
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I'm sorry but I dont understand what you mean by the independence of e^ix and e^x.
 
  • #7
dextercioby
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[tex] Ae^{ix}+Be^{x}=0\Leftrightarrow A=B=0 [/tex]


Daniel.
 

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