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Math Help - [SOLVED] proof for complx/trig eqn?

  1. #1
    crb
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    [SOLVED] proof for complx/trig eqn?

    plz have a look at the aatached picture
    thank you
    Attached Thumbnails Attached Thumbnails [SOLVED] proof for complx/trig eqn?-5_page_5.jpg  
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  2. #2
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     sinh(z) = \frac{e^{z} - e^{-z}}{2}

      = \frac{e^{x+iy} - e^{-x-iy}}{2}

      = \frac{e^{x}.e^{iy} - e^{-x}.e^{-iy}}{2}

      = \frac{e^{x}.e^{iy} - e^{-x}.\frac{1}{e^{iy}}}{2}

      = \frac{e^{x}.(cos(y)+isin(y)) - \frac{1}{e^{x}.(cos(y)+isin(y))}}{2}
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  3. #3
    crb
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    and how about the conjugate of sinh(z)....??
    how do i get sinh(z) = some u + iv form????
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  4. #4
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    Quote Originally Posted by crb View Post
    and how about the conjugate of sinh(z)....??
    how do i get sinh(z) = some u + iv form????
      = \frac{e^{x}.(cos(y)+isin(y)) - \frac{1}{e^{x}.(cos(y)+isin(y))}}{2}

    Here you have a complex number which is composed of subtraction of two complex numbers.

    Remember that if  w and z are complex numbers then:

     \overline{(v - w)} = \overline{v} - \overline{w}

    In this case  v = e^{x}.cos(y)+ie^{x}sin(y)  w = \frac{1}{e^{x}.cos(y)+ie^xsin(y)}

    And you're trying to find \frac{1}{2} \overline{(v - w)}

    Get w in a nicer form by multiplying top and bottom by the conjugate of the denominator first.

     w = \frac{1}{e^{x}.cos(y)+ie^xsin(y)}  = \frac{e^{x}.cos(y)-ie^xsin(y)}{e^{2x}.cos^2(y)+ie^{2x}sin^2(y)} =   \frac{cos(y)-isin(y)}{e^{x}}

    Hence

    \frac{1}{2}( v - w) = \frac{1}{2}(e^{x}.cos(y)+ie^{x}sin(y) - (\frac{cos(y)-isin(y)}{e^{x}})

    Should be relatively easy to find the conjugate now.
    Last edited by Mush; January 7th 2009 at 01:02 PM.
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  5. #5
    crb
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    Thank u very much...just managed to prove it with ur help..thank you
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