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Math Help - principal vaue of a complex number

  1. #1
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    principal vaue of a complex number

    I strugged for half an hour trying to find materials for this problem.

    (3+j4)^{1+2j}


    could you lead me to a good study material for this topic ?

    thanks
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  2. #2
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    Is this \displaystyle (3+4j)^{1+2j}?

    If so, you should note that complex exponentiation is best done using the exponential form of the complex number.

    \displaystyle |3+4j| = \sqrt{3^2 + 4^2} = 5

    \displaystyle \arg{(3+4j)} = \arctan{\frac{4}{3}}.


    So \displaystyle 3 + 4j = 5e^{j\arctan{\frac{4}{3}}}


    \displaystyle (3 + 4j)^{1+2j} = (5e^{j\arctan{\frac{4}{3}}})^{1+2j}

    \displaystyle = 5^{1+2j}\,e^{j\arctan{\frac{4}{3}}(1 + 2j)}

    \displaystyle = e^{\log{(5^{1+2j})}}\,e^{j\arctan{\frac{4}{3}}(1+2  j)}

    \displaystyle = e^{(1+2j)\log{5}}\,e^{j\arctan{\frac{4}{3}}(1+2j)}

    \displaystyle = e^{(1+2j)\log{5}+j\arctan{\frac{4}{3}}(1+2j)}

    \displaystyle = e^{\log{5} + 2\log{5}j + j\arctan{\frac{4}{3} - 2\arctan{\frac{4}{3}}}

    \displaystyle = e^{\log{5} - 2\arctan{\frac{4}{3}} + \left(2\log{5} + \arctan{\frac{4}{3}}\right)j}

    \displaystyle = e^{\log{5}-2\arctan{\frac{4}{3}}}\,e^{\left(2\log{5} + \arctan{\frac{4}{3}}\right)j}

    \displaystyle = e^{\log{5}- 2\arctan{\frac{4}{3}}}\left[\cos{\left(2\log{5} + \arctan{\frac{4}{3}}\right)} + i\sin{\left(2\log{5} + \arctan{\frac{4}{3}} \right)}\right]

    \displaystyle = e^{\log{5} - 2\arctan{\frac{4}{3}}}\cos{\left(2\log{5} + \arctan{\frac{4}{3}}\right)} + i\,e^{\log{5} - 2\arctan{\frac{4}{3}}}\sin{\left(2\log{5} + \arctan{\frac{4}{3}}\right)}

    \displaystyle = 5e^{-2\arctan{\frac{4}{3}}}\cos{\left(2\log{5} + \arctan{\frac{4}{3}}\right)} + 5i\,e^{-2\arctan{\frac{4}{3}}}\sin{\left(2\log{5} + \arctan{\frac{4}{3}}\right)}




    PHEW!!!
    Last edited by Prove It; December 18th 2010 at 02:48 AM.
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  3. #3
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    wow. never thought that it's that complicated. thank you !

    the source where i got it from says that the answer is -0.42 - j0.66

    also, my python calculator also says

    >>> pow((3+4j),(1+2j))
    (-0.41981317556195746-0.6604516942073323j)

    how can we prove it ?
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  4. #4
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    Why don't you try evaluating the real and imaginary parts of the answer I've posted?
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