- #1
spaderdabomb
- 49
- 0
I was doing some math, typed in (eix)(e-ix) and it came out as 1.
I was expecting it to come out as simply ex. Explain pleaseeee thanks =).
I was expecting it to come out as simply ex. Explain pleaseeee thanks =).
Eval said:And for those that didn't catch the simpler nature of the problem, as dextercioby said, you can use basic laws of exponents:
eixe-ix=eix-ix=e0=1.
Alternatively:
eixe-ix=eix/eix=1.
:)
Dickfore said:Well, micromass, actually there is a BIG reason why the law of exponents should apply to complex numbers. It's because the exp(z) is an analytic continuation of the natural exponent ex that preserves the functional identity:
[tex]
f(z_1 + z_2) = f(z_1) f(z_2)
[/tex]
Dickfore said:Sure, but show me a proof of Euler's identity that was used in post #4!
Dickfore said:So, then, poster #2 was not wrong.
Dickfore said:Sure, but show me a proof of Euler's identity that was used in post #4!
Eval said:
Since that is a valid representation of eu, you can see how it leads to representing an imaginary power of e directly to a function of cosine and sine. I think this is how Euler showed the identity, too.Dickfore said:Why did you define the exponential through the Maclaurin series in step 1?
Eval said:Since that is a valid representation of eu,
The symbol "e" in this equation represents the mathematical constant known as Euler's number, which has a value of approximately 2.71828. It is a fundamental constant in mathematics and is used to represent exponential growth.
When the exponent is a complex number, the result of raising e to that power will also be a complex number. This is because complex numbers have both a real and an imaginary component, and e raised to a complex power involves both of these components.
This is due to the properties of exponents, specifically the property that states that when two numbers with the same base are multiplied, their exponents are added. In this case, the exponent of e in (e^ix)(e^-ix) is ix + (-ix) = 0. Any number raised to the power of 0 is equal to 1, so the result is 1.
Yes, this equation can be applied to any value of x, whether it is a real number or a complex number. The result will always be 1.
This equation is commonly used in complex analysis and physics, specifically in quantum mechanics. It is also used in mathematical proofs and calculations involving complex numbers and exponential functions.