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What is the digamma function?

Can somebody tell me in a simple language what is the digamma function and its application in problem solving..I understand the gamma function and its uses but i am not able to understand digamma.wikipedia seems to be too elaborate and intricate in explanantion.Any help would be much appreciated.

Note by Incredible Mind
2 years, 4 months ago

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You can read about it here Kishlaya Jaiswal · 2 years, 1 month ago

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Sorry that I'm late, but I see nobody has commented yet! I hope I can be of service. If you are familiar with the gamma function, then I assume you are familiar with differentiation as well. Simply put, the digamma function is the derivative of the gamma function, divided by the gamma function. That is, \[\psi (x) = \frac{\Gamma '(x)}{\Gamma (x)}.\] At first sight it does not seem like much, but it has proven infinitely useful in statistics, calculus, and analysis.
I cannot think of any applications that I can explain with detail in a single comment, but a few applications are regularizing a divergent sum (kind of like finding a "theoretical" value of a divergent sum), estimating values of the gamma function efficiently and accurately, and estimation of maximum likelihood of a gamma distribution in statistics.
By itself it is not a particularly meaningful function like the gamma function is. However, studying it has saved a great deal of time in calculation. For that reason I like to think of digamma as a mathematical tool. A hammer by itself is not interesting, but when you use it to secure a nail, you remember how useful a hammer actually is. As such, the hammer becomes interesting. The digamma function is like the hammer in that sense: it is useful in shaping estimations and inequalities, thereby making it worthy of study. Caleb Townsend · 2 years, 3 months ago

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@Caleb Townsend sir can u please explain where it can be useful with a simple example like direct estimates for otherwise complex integrals...such and such. Thank you for posting this. Incredible Mind · 2 years, 3 months ago

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@Incredible Mind Yes, here's the simplest example I can think of, although this is not an approximation. Consider the convergent integral \[\int_0^{\infty}[\frac{1}{te^t} - \frac{1}{e^t-1}] dt\] How does one go about solving this? You might consider writing it as the difference of two integrals (namely of \(1/(te^t)\) and \(1/(e^t-1)\)) but you will see these are both divergent. So there are a few other methods that come to mind, but by far the most efficient is noticing that it is equal to \(\psi(1)\) since this is one of the digamma function's many integral representations. We know \(\psi(1) = -\gamma \approx -.57722\) so therefore \[\int_0^{\infty}[\frac{1}{te^t} - \frac{1}{e^t-1}] dt \approx -.57722\] The digamma function can be written as logarithmic or exponential integrals in many ways, this is but one way. If you find an integral or sum that can be approximated with any of the integral representations, it greatly simplifies calculations. You can find other integral representations online to get an idea of what the digamma can be used for in calculus. Caleb Townsend · 2 years, 3 months ago

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@Caleb Townsend i am starting to get it.thank you Incredible Mind · 2 years, 3 months ago

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