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Math Help - Transforming a complex exponentiation into a complex rectangular notation

  1. #1
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    Transforming a complex exponentiation into a complex rectangular notation

    The following exponentiation:

    3e^(1+pi i)

    Needs to be written in rectangular notation:

    a + ib

    Where "i" is the imaginary unit.

    This is what I've found out myself so far:

    I know that in order to find "a" and "b" I can use the argument "v" in the exponential notation:

    re^(iv)

    a = cos(v)
    b = sin(v)

    I also know that e^(iv1+iv2) = e^(iv1) * e^(iv2)

    So if I try to apply these rules I get:

    3e^(1+pi i) = 3e^1 * 3e^pi

    Now calculating "a" I get:

    a = 3*cos(1) * 3*cos(pi) = 3*cos(1) * -3

    And for "b":

    b = 3*sin(1) * 3*sin(pi) = 0

    But I get the feeling that 3*cos(1) is wrong.
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  2. #2
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    Re: Transforming a complex exponentiation into a complex rectangular notation

    e^{1} \ne \cos(1)

    e^{1+i\pi}\;=\;e^{1}\cdot e^{i\pi}\;=\;e\cdot (\cos(\pi)+i\cdot\sin(\pi))\;=\;e\cdot (-1 + 0)\;=\;-e
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  3. #3
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    Re: Transforming a complex exponentiation into a complex rectangular notation

    So e^1 simply equals e?
    Does that mean that 3e^1 simply equals 3e?

    And shouldn't it be 3*cos(pi) and 3*sin(pi)?
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  4. #4
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    Re: Transforming a complex exponentiation into a complex rectangular notation

    Yes. That "i" is more important than you were thinking. Without it, it's just Real Numbers.
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    Re: Transforming a complex exponentiation into a complex rectangular notation

    But here is what I don't understand:

    3e^1 = 3e
    3e^pi i = 3*cos(pi) + i 3*sin(pi)

    3e * (3*cos(pi) + i 3*sin(pi)) = 3e * (-3+0) = -9e

    But my facit says "-3e"
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  6. #6
    MHF Contributor Siron's Avatar
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    Re: Transforming a complex exponentiation into a complex rectangular notation

    Note that:
    3e^{1+i\pi}\neq 3e\cdot 3e^{i\pi}
    3e^{1+i\pi}=3e\cdot e^{i\pi}
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