How do you differentiate the incomplete gamma function?

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Exactly what the title says. Not an easy function to work with, for some reason it doesn't have a known indefinite integral even though it seems like it almost should. The answer seems to have something to do with the polygamma function though the exact process of differentiating the gamma function seems non-existent on the internet.

Edited by MWresearch
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The incomplete gamaa function is a function of 2 variables. The formal derivative with respect to either variable looks easy enough. I think you should elaborate on what you are questioning.

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The incomplete gamaa function is a function of 2 variables. The formal derivative with respect to either variable looks easy enough. I think you should elaborate on what you are questioning.

The first problem is d/dx*gamma(x)=gamma(x)*polygamma_0(x) and the polygamma function is already defined in terms of the original gamma function and the derivative of the gamma function, the logic is completely circular, all anyone is saying by writing that is that the derivitive of the gamma function is the derivative of the log of the gamma function divided by the gamma function, it explains nothing about what the actual polygamma function is and what combination of operators it performs on an input and thus there is no clear way to differentiate it. It's like saying that the integral of (2/sqrt(pi))e^(-x^2) is the error function when the error function is only defined as the integral of (2/sqrt(pi))e^(-x^2), again, the logic is circular and absolutely zero known functions are represented in closed form after taking the integral of the Gaussian curve. The polygamma function appears to be the exact same type of ambiguous problem except with derivatives instead of integrals. Why can't they just admit they don't know the derivative? I might as well say the derivative of the gamma function is spaghetti where spaghetti = the derivative of the gamma function, saying that solves nothing and shows nothing about taking the actual derivative of the function because the logic is simply circular, so if I don't know what the derivative is, how am I suppose to differentiate it?

And THEN holding the input of the gamma function fixed while varying the boundary is suppose to somehow be x^(t-1)e^(-x) except there's absolutely no known elementary function that you can use to evaluate the boundaries of the lower incomplete (or upper) gamma function at x and infinity to verify that result after integrating the original function, again, completely circular logic that solves nothing. If they don't know of any such function or combination of operators that can accurately describe the integral of the original function in closed form they should simply admit it and stop wasting everyone's time. But since they won't tell anyone and make up this circular logic crap, no outside viewer can have any way of knowing.

Edited by MWresearch
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$\Gamma (s,x)=\int_x^\infty t^{s-1}e^{-t}dt$

$\gamma (s,x)=\int_0^x t^{s-1}e^{-t}dt$

I don't see any problem with getting derivatives.

Edited by mathematic

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