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Math Help - complex number, horizontal strip

  1. #1
    Junior Member
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    complex number, horizontal strip

    For a fixed complex number \lambda, show that if m and n are large integers, then the equation e^z=z+\lambda has exactly m+n solutions in the horizontal strip \{ -2\pi i m < \text{Im}(z)<2 \pi i n \}.


    The hint says to consider f(z)=e^z -z-\lambda and sketch the image of the boundary of \{  -R< x<R, -2 \pi m < y < 2 \pi n \}. Keep in mind |\lambda | << m, n << R.


    I do not see how to prove this now. I need some help on this one. Thank you.
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  2. #2
    Super Member
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    I'd like to try and tackle this if I may although not via Rouche': First, it's easy to solve for the roots of f(z)=e^z-z-\lambda in terms of the Lambert-W function:

    z_n=-\lambda-\textbf{W}\left(-e^{-\lambda}\right)

    and keep in mind the W-function is infinitely valued and looks a bit like the log multi-sheet so we should actually write it as:

    z_n=-\lambda-\textbf{W}\left(n,-e^{-\lambda}\right)

    where n is the sheet number. The top left plot is the contour for n=2, m=1, R=4, \lambda=1+i and the zeros z_n as the black dots. As you can see, the contour is enclosing three zeros. The lower-left is the image of f(z) over that contour. On the brown contour we have: e^{R+it}-(R+it)-(1+i) and if R is large, the expression is dominated by e^{R+it} giving rise to the circular contours around the origin but note the remaining contours complete the circuit around the origin giving rise to a winding number of n+m=3. This is primarily due I think to the blue contour over which we have e^{-R+iy}-(-R+iy)-(a+ib) and since the e^{-R} term is small, the real part is dominated by R-a and if this is positive, then the circular contour complete the circuit around the origin n+m times as shown by the lower left plot. However, in the case of R=4 and \lambda=7+i as the second set shows, then R-a=-3 which causes a detour around the origin and gives rise to a winding number of n+m-1=2. This is reflected in the top right plot showing only two zeros enclosed by the contour . . . I'm thinkin' B at best.
    Attached Thumbnails Attached Thumbnails complex number, horizontal strip-argument-problem.jpg  
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