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Math Help - Consider the complex number

  1. #1
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    Consider the complex number

    a)Consider the complex number z1=-3+4i

    i) Express this number as z1=re^iθ

    ii) Find ln(z1)


    SO WE KNOW WHAT r=5

    AND WE KNOW THAT θ=126,... ?

    here i have a problem - i dont know how to slove the a angle (this angle must be with PI) having the angle i know how to but to z1=re^iθ


    I dont know how to solve ii) i found on wikipedia the rule but i dont understnd - can somebody help ?
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  2. #2
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    Er you just convert the degree to radian.

    We know that 180 degrees correspond to pi radian.

    hence 180==pi rad
    so 126 degree will correspond to 126pi/180 radian

    so your angle is essentially (63/90)pi radian or 0.7pi

    so the euler expression is merely 5e^(0.7pi)
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  3. #3
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    Ah ok what about secend question ?
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  4. #4
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    The definition of a logarithm of a complex number is given by

    ln(z1)= ln(r)+ i(theta)

    so ln(z1) is simply ln(5)+i(0.7pi)

    The result is very simple to obtain

    as z1=re^iθ
    Taking logarithms to the base "e"(i.e. "ln") on both sides,
    ln(z1)=ln{5(e^iθ )}
    Using the property of logarithms,

    ln(z1)= ln(5) + ln(e^iθ)
    =ln(5) + iθln(e)
    but ln(e)=1
    so ln(z1)=ln5 + iθ
    just plug in theta.
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  5. #5
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    WOW you are a mathematical genius thanks for help
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