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Functional analysis (?)

  1. Aug 18, 2013 #1
    Hello,

    While analysing the asymptotic value of a ratio of a bessel and a hankel function, I reduced it to something of the form

    [(1 + β/n)^ n * (1 + n/β)^ β] / 2^(n+β) ; n and β are integers and greater than 1

    how do I show that the above expression is always less than 1, for n≠β. When n=β, the above expression becomes equal to 1.

    Or relatedly, if I have to find the line of maximum for a 2D expression given above (for varying n and β), how do I go about ?

    Thanks!
     
  2. jcsd
  3. Aug 18, 2013 #2

    verty

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    I've played around with this a little. This formula is symmetric in β,n with two variables but as mentioned, β = n is enough to give the maximum value.

    Let x = 1 + n/β, y = n + β.
    Thus ##\frac{y}{x} = β, \frac{y(x-1)}{x} = n## and x > 1, y ≥ 2.

    The formula simplifies to
    ##[ \frac{1}{2} x (x-1)^{\frac{1}{x} - 1} ]^y##

    Y is irrelevant here, it won't affect which x gives the maximum. Discarding y, the derivative of what remains has numerator
    ##-(x-1)^{\frac{1}{x}} ln(x-1)##

    The exponential part is never 0, therefore x = 2 is the only stationary point. I hope this is not the best way to show this.
     
  4. Aug 19, 2013 #3

    dextercioby

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    Your question is from calculus, or mathematical analysis if you prefer. Functional analysis is built on point set topology and is an abstractization of calculus.
     
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