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    So I've been thinking a lot lately about the graph of y = (-1)^x. I'm in year 13 so as you'd assume, I can't do much other than cover special cases.

    If x is integer: If x is odd, y = -1
    If x is even, y = 1
    If x is rational:
    Let n represent any number.
    If x can be expressed as n / 2n + 1 then:
    If n is even, y = 1
    If n is odd, y = -1

    So basically, the only cases that work in the real planes are when x can be expressed as a rational number with an odd denominator... Is this it?

    I asked my Further Maths teacher and he said go research the gamma function, but that didn't seem to answer any questions. Do I need to research further?

    Any info on this graph would be helpful, thank you!
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    (Original post by ComputerMaths97)
    So I've been thinking a lot lately about the graph of y = (-1)^x. I'm in year 13 so as you'd assume, I can't do much other than cover special cases.

    If x is integer: If x is odd, y = -1
    If x is even, y = 1
    If x is rational:
    Let n represent any number.
    If x can be expressed as n / 2n + 1 then:
    If n is even, y = 1
    If n is odd, y = -1

    So basically, the only cases that work in the real planes are when x can be expressed as a rational number with an odd denominator... Is this it?

    I asked my Further Maths teacher and he said go research the gamma function, but that didn't seem to answer any questions. Do I need to research further?

    Any info on this graph would be helpful, thank you!
    You might find it easier to think through this if you remember the identity

     a^{x} = \exp(x \ln(a))

    This of necessity takes you into the complex plane (ln(-1) = i Pi) but will allow you to investigate things systematically.

    In fact, if f(z) = (-1)^{z}, then \ln(f(z)) = z \ln(-1) and therefore      f(z) = \exp(i \pi z)

    which is probably a lot easier to think about!
 
 
 
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