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- arindameducationusc

Can anyone explain x^pi?

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- arindameducationusc

Can anyone explain x^pi?

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- ParthKohli

It's \(x\) raised to the power \(\pi\).

- arindameducationusc

yes, i will upload the graph... I have a doubt. wait..

- arindameducationusc

Ya this one

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- arindameducationusc

The imaginary part and real part...

- arindameducationusc

Can you explain...

- ParthKohli

For a positive number, this is of course a purely real number. Thus the imaginary part is zero.

- ParthKohli

You do understand complex numbers, right?

- arindameducationusc

Yes I understand Complex numbers. So that means we can graph complex numbers? I didn't know that!

- ParthKohli

Yeah. For negative \(x\), this function has complex values.

- arindameducationusc

Do you have any reference in which I can study complex number graphs, any good books or video link?

- Astrophysics

http://www.mathsisfun.com/numbers/complex-numbers.html

- Astrophysics

khanacademy is awesome for all math

- Astrophysics

as well

- arindameducationusc

Then if x=3, then 3^3.14 should be real, but in graph it is imaginary..

- anonymous

Its just x^3.14

- zepdrix

@arindameducationusc You'll notice that the orange line stays on the x-axis when x is positive.
The imaginary part is zero for all positive x. So there is no imaginary part over there.

- arindameducationusc

- phi

you may know
\[ e^{i\pi}= \cos\pi + i \sin \pi = -1 + 0 \ i = -1 \]
when x is negative, x^pi is the same as
\[ (- |x|)^\pi = \left( e^{i\pi} |x|\right)^\pi \\= e^{i\pi^2} |x|^\pi=
(\cos\pi^2 + i \sin \pi^2)|x|^\pi \\
(-x)^\pi \cos\pi^2 + i (-x)^\pi \sin \pi^2
\]
which has a non-zero imaginary component

- Astrophysics

^ Yes, that's very useful, Euler's equation

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