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Integral equation for Xi-function

  1. Nov 17, 2009 #1
    think i have discovered an integral equation for the Xi-function

    [tex] \Xi (z)= A\int_{-\infty}^{\infty} \phi (x/2)\Xi(x).\Xi(x+z) \frac{dx}{x} [/tex]


    [tex] \Phi(u) = \sum_{n=1}^{\infty}(2\pi ^{2} n^{4}e^{9u}-3\pi n^{2}e^{5u} )exp(-\pi n^{2}e^{4u}) [/tex]

    and 'A' is a Real constant.
  2. jcsd
  3. Nov 17, 2009 #2
    of course i am referring to [tex] \Xi(z)= \xi (1/2+iz) [/tex] and due to the functional equation this Xi is even so we can formulate the integrale equation as

    [tex] \Xi (z)= A\int_{-\infty}^{\infty} \phi (x/2)\Xi(x).\Xi(x+z) \frac{dx}{x}=A\int_{-\infty}^{\infty} \phi (x/2)\Xi(x).\Xi(x-z) \frac{dx}{x} [/tex]
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