Problem 46025. Evaluate the gamma function

Created by ChrisR

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<div style = "text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 93.3333px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407.5px 46.6667px; transform-origin: 407.5px 46.6667px; vertical-align: baseline; "><div style="block-size: 63.3333px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 31.6667px; text-align: left; transform-origin: 384.5px 31.6667px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.0583px 7.66667px; transform-origin: 12.0583px 7.66667px; unicode-bidi: normal; "><span style="">The</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.66667px; transform-origin: 1.94167px 7.66667px; unicode-bidi: normal; "><span style=""> </span></span><a target='_blank' href = "https://mathworld.wolfram.com/GammaFunction.html"><span style=""><span style="">gamma function</span></span></a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 302.642px 7.66667px; transform-origin: 302.642px 7.66667px; unicode-bidi: normal; "><span style=""> is a generalization of the factorial, and it appears in many applications such as evaluating certain integrals, working with probability distributions, and evaluating fractional derivatives. MATLAB includes the function</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 7.66667px; transform-origin: 1.94167px 7.66667px; unicode-bidi: normal; "><span style=""> </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 19.25px 7.66667px; transform-origin: 19.25px 7.66667px; unicode-bidi: normal; "><span style="font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; perspective-origin: 19.25px 8px; transform-origin: 19.25px 8px; ">gamma</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.66667px; transform-origin: 3.88333px 7.66667px; unicode-bidi: normal; "><span style="">, but it accepts only real arguments.</span></span></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 232.467px 7.66667px; transform-origin: 232.467px 7.66667px; unicode-bidi: normal; "><span style="">Write a function that evaluates the gamma function for complex arguments.</span></span></div></div></div>

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Last Solution submitted on Jun 03, 2023

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Rafael S.T. Vieira on 24 Oct 2020

Several formulas can approximate the gamma function on the complex domain, but all of them have issues. Please, look at its plot before starting. Nonetheless, there is a trick that makes them all work; and you must find it (I've lost some time trying different formulas, adding more terms, but it is not the way to go).

Rafael S.T. Vieira on 24 Oct 2020

I've found a 1968-paper by Wrench, Concerning Two Series for the Gamma Function, which gives us the 20th first terms of one formula for instance. https://www.ams.org/journals/mcom/1968-22-103/S0025-5718-1968-0237078-4/ (It would be more than enough if there weren't issues with all approximations.)

Nikolaos Nikolaou on 24 Dec 2021

lol that was fun xD

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Problem 46025. Evaluate the gamma function

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Problem 46025. Evaluate the gamma function

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