Now, if you’ve ever taken Calculus I (or even just watched “A Beautiful Mind” with Russell Crowe), you know that calculating derivatives can be a real pain in the butt. But no need to get all worked up! With this handy-dandy formula, we can calculate successive derivatives of the gamma function like it’s nobody’s business.

So what exactly is T(m,s,x)? Well, let me break it down for you:

T(m,s,z) = (1-m)^(-1) * (m-2)! * [Gamma(s-t) * z^(t-1)] | t=0 + sum from n=0 to infinity of (-1)^n * n! * (s+n)^(-m)

Now, I know what you’re thinking “That looks like a bunch of gibberish!” And you’d be right. But trust me, it’s actually pretty simple once you break it down into smaller parts.

First off, the variables: m, s, and x are all just numbers (or variables) that we can plug in to get our desired result. z is a little bit different it represents a complex number (i.e., a number with both real and imaginary parts).

So what does this formula actually do? Well, let’s say you want to calculate the third derivative of the gamma function at x=2. To do that, we would plug in m=3, s=x=2 into our fancy T(m,s,z) formula:

T(3,2,2) = (1-3)^(-1) * (3-2)! * [Gamma(2-t) * 2^(t-1)] | t=0 + sum from n=0 to infinity of (-1)^n * n! * (2+n)^(-3)

Now, I know what you’re thinking “That looks like a bunch of gibberish!” And you’d be right. But trust me, it’s actually pretty simple once you break it down into smaller parts.

First off, the variables: m, s, and x are all just numbers (or variables) that we can plug in to get our desired result. z is a little bit different it represents a complex number (i.e., a number with both real and imaginary parts).

So what does this formula actually do? Well, let’s say you want to calculate the third derivative of the gamma function at x=2. To do that, we would plug in m=3, s=x=2 into our fancy T(m,s,z) formula:

T(3,2,2) = (1-3)^(-1) * (3-2)! * [Gamma(2-t) * 2^(t-1)] | t=0 + sum from n=0 to infinity of (-1)^n * n! * (2+n)^(-3)

Now, let’s break that down into smaller parts:

First off, we have the factorial function (represented by !). This is just a fancy way of saying “multiply all the numbers from 1 to n”. So in our case, (3-2)! would be equal to 1 * 2 = 2.

Next up, we have the Gamma function (represented by Gamma(s)). This is a mathematical function that extends the factorial function to non-integer values of s. So in our case, Gamma(2-t) would be equal to something like this:

Gamma(2-t) = (1-t)^(-2) * [integral from x=0 to infinity of e^(-x) * x^(s-1)] dx

Now, the summation part. This is where we add up all those ***** (-1)^n * n! terms. The idea here is that as n gets larger and larger (i.e., goes to infinity), these terms will eventually cancel each other out and leave us with a nice, clean result.

Of course, this is just scratching the surface there’s so much more to learn about math and science than we could ever possibly cover in one article.

So go out there and start exploring! Who knows maybe someday you’ll be the next Russell Crowe, solving complex equations and saving lives in the process.