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

• pt176900
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.
pt176900
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?

HINT:make x=0.

Daniel.

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

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?

Nope.That x=0 will probe that A+B=C.Now u'll have to make use of the independence of the $e^{ix}$ and $e^{x}$ for $x\in \mathbb {R}$.

Daniel.

I'm sorry but I don't understand what you mean by the independence of e^ix and e^x.

$$Ae^{ix}+Be^{x}=0\Leftrightarrow A=B=0$$

Daniel.

## 1. What is a proof in science?

A proof in science is a logical and systematic demonstration of the validity of a scientific idea or theory. It involves using evidence, experimentation, and reasoning to support a hypothesis or claim.

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To construct a proof, you need to clearly define your hypothesis or claim, gather relevant evidence and data, and use logical reasoning to connect your evidence to your hypothesis. It is important to carefully document each step and make sure your argument is sound and supported by evidence.

## 3. What makes a proof valid?

A proof is valid when it follows a logical and systematic approach, is supported by evidence, and can be replicated and verified by others. A valid proof should also adhere to the scientific method and be free from any biases or errors.

## 4. How can I improve my proof?

To improve your proof, you can seek feedback from other scientists or experts in the field. You can also conduct further experiments or gather more evidence to strengthen your argument. It is important to be open to criticism and continuously refine your proof until it is supported by solid evidence.

## 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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