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Gama function solves f'=f?

  1. Jun 15, 2010 #1
    I have tried to calculate the derivative of Gamma function and got a strange result, which
    is obviously wrong. Can someone find the mistake?

    Definition:
    Gamma[z]=Integral[t^(z-1)exp(-t)dt]

    Derivative:
    (d/dz)Gamma[z]=Integral[(d/dz)t^(z-1)exp(-t)dt]=Integral[(z-1)t^(z-2)exp(-t)dt]=
    (z-1)*Gamma[z-1]=Gamma[z]

    Looks like gamma solves the equation f'=f, but this can't be true, since only
    exponential function solves this equation.
     
  2. jcsd
  3. Jun 15, 2010 #2
    Your problem is with (d/dz)t^(z-1). What you computed is really (d/dt)t^(z-1). You need to use logarithmic differentiation: (d/dz)t^(z-1) = t^(z-1) ln t
     
  4. Jun 15, 2010 #3
    You are right, I derived with respect to the wrong variable. I wanted to calculate the derivative of gamma at least at integer parameter values, but it seems I won't be able to do this, since I can't calculate the resulting integral.
     
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