How do I solve this integral

[VictorFreg]

I'm stuck on this:
\(\displaystyle \int_0^\infty log(\mu z/\phi) \frac{1}{\Gamma (\phi)}z^{\phi-1} e^{-z} dz \)

If I'm not wrong on the statistics part of the problem, it should be \(\displaystyle \frac{\Gamma'(\phi)}{\Gamma(\phi)} + log(u) - log(\phi) \), but I can't manage the product inside the integral.

 

[Deleted member 4993]

I'm stuck on this:

integral from 0 to infinity: log(uz/h) * 1/Gamma(h) * z^(h-1) * exp(-z) dz

If I'm not wrong on the statistics part of the problem, it should be gamma'(h)/gamma(h) + log(u) - log(h), but I can't manage the product inside the integral.

Is it:

\(\displaystyle \displaystyle{\int_0^\infty \left [ Log\left (\frac{uz}{\phi}\right )\right ] * \frac{1}{\Gamma (\phi)} * z^{(\phi -1)} * e^{-z} dz}\)

Why don't you differentiate the given answer to see if you get the integrand back?

 

[VictorFreg]

I figured how to use tex on the text now to make it clearer.

It's part of a proof, so I need to show that'll be the result somehow.

 

[daon2]

I'm stuck on this:
\(\displaystyle \int_0^\infty log(\mu z/\phi) \frac{1}{\Gamma (\phi)}z^{\phi-1} e^{-z} dz \)

If I'm not wrong on the statistics part of the problem, it should be \(\displaystyle \frac{\Gamma'(\phi)}{\Gamma(\phi)} + log(u) - log(\phi) \), but I can't manage the product inside the integral.

What is the "statistics part of the problem"?

Have you tried integration by parts? Maybe it will help you to know

\(\displaystyle \displaystyle \Gamma(t):=\int_0^{\infty} z^{t-1} e^{-z} dz\)

Although to me, the resulting integral does not appear any easier, maybe you have results from your class which make it doable.

 

[Ishuda]

I'm stuck on this:
\(\displaystyle \int_0^\infty log(\mu z/\phi) \frac{1}{\Gamma (\phi)}z^{\phi-1} e^{-z} dz \)

If I'm not wrong on the statistics part of the problem, it should be \(\displaystyle \frac{\Gamma'(\phi)}{\Gamma(\phi)} + log(u) - log(\phi) \), but I can't manage the product inside the integral.

No Idea but you might look at
http://www.wolframalpha.com/input/?i=integrate+log%28u+z%2Fp%29+z^%28p-1%29+e^%28-z%29+dz