A conflicting result of Euler's formula

[Undoubtedly0]

It can be shown from Euler's formula that

$$\left(-1\right)^x = \cos(\pi x) + i\sin(\pi x)$$

However, consider $x = 2/3$. The left expression gives

$$\left(-1\right)^\frac{2}{3} = \left(\left(-1\right)^2\right)^\frac{1}{3} = \left(1\right)^\frac{1}{3} = 1$$

while the right expression gives

$$\cos\left(\frac{2\pi}{3}\right) + i\sin\left(\frac{2\pi}{3}\right) = -\frac{1}{2} + \frac{\sqrt{3}}{2}i$$

What has gone wrong here? Thanks all.

 

[andrien]

because you have not done it in general.
(-1)^x=cos(n*pi*x)+isin(n*pi*x) where n is odd integer.now put x=2/3 and n=3.
you get it equal to 1.also check out that -1 has three cube roots so just check out that that complex number is arising because of other roots.

 

[DonAntonio]

It can be shown from Euler's formula that

$$\left(-1\right)^x = \cos(\pi x) + i\sin(\pi x)$$

However, consider $x = 2/3$. The left expression gives

$$\left(-1\right)^\frac{2}{3} = \left(\left(-1\right)^2\right)^\frac{1}{3} = \left(1\right)^\frac{1}{3} = 1$$

while the right expression gives

$$\cos\left(\frac{2\pi}{3}\right) + i\sin\left(\frac{2\pi}{3}\right) = -\frac{1}{2} + \frac{\sqrt{3}}{2}i$$

What has gone wrong here? Thanks all.

Complexpowers, roots don't behave as real ones. Read about multivaluate complex functions...or wait until the appropiate course at university.

DonAntonio

 

[Undoubtedly0]

because you have not done it in general.
(-1)^x=cos(n*pi*x)+isin(n*pi*x) where n is odd integer.now put x=2/3 and n=3.
you get it equal to 1.also check out that -1 has three cube roots so just check out that that complex number is arising because of other roots.

How does one derive $(-1)^x=\cos(n\pi x)+i\sin(n\pi x)$?

To derive $(-1)^x=\cos(\pi x)+i\sin(\pi x)$ I simply used
$$\ln\left((-1)^x)\right)=x\ln(-1)=x\pi i$$
therefore $(-1)^x = e^{x\pi i} = \cos(\pi x)+i\sin(\pi x)$.

Where did I miss the general case? Thanks again.

 

[andrien]

can't you just verify that -1 is equal to exp(i*n*pi) when n is odd.

 

[Techian]

The equation, (-1)^x=cos(pi*x)+i*sin(pi*x) is derived from Euler's identity based on the assumption that 'x' is an integer. Hence substituting fractional value for 'x' in the equation will lead to erroneous results. The derivation is as follows,
e^(i*y)=cos(y)+i*sin(y)...(Original Euler's identity)
Now replacing 'y' with 'pi*x', you will end up with
e^(i*pi*x)=cos(pi*x)+i*sin(pi*x)...(1)
Assuming that 'x' is an integer, cos(pi*x)=(-1)^x and sin(pi*x)=0.
substituting them in (1)
e^(i*pi*x)=(-1)^x...(2)
substituting (2) in (1), you will end up with (-1)^x=cos(pi*x)+i*sin(pi*x) for x is an integer.