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Title: Gamma Function
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Series: Advent of Mathematical Symbols
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YouTube-Title: Gamma Function
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Bright video: https://youtu.be/5uLxCQrgPkU
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Dark video: https://youtu.be/tvSov96Odgk
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Ad-free video: Watch Vimeo video
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Print-PDF: Download printable PDF version
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: aoms05_sub_eng.srt
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Timestamps (n/a)
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Subtitle in English
1 00:00:00,729 –> 00:00:04,571 The mathematical Symbol of today is the so called Gamma function.
2 00:00:04,686 –> 00:00:06,309 Written with a capital Gamma.
3 00:00:07,429 –> 00:00:11,709 and the name of the variable we put in is usually given by z.
4 00:00:12,443 –> 00:00:18,077 Now the explicit definition of the Gamma function is not so simple, because it’s given by an integral.
5 00:00:19,014 –> 00:00:22,408 Indeed we integrate from 0 to infinity.
6 00:00:23,171 –> 00:00:27,033 and the variable for the integration is given by x.
7 00:00:27,800 –> 00:00:32,706 and the number z from the left hand side we find here in the exponent of x.
8 00:00:33,614 –> 00:00:36,663 Because it’s x to the power (z-1).
9 00:00:37,214 –> 00:00:41,794 Moreover this is then multiplied with e to the power (-x).
10 00:00:42,686 –> 00:00:48,471 So you see this is not a simple definition for a function and to make it even more complicated
11 00:00:48,472 –> 00:00:52,357 i can tell you this z could even be a complex number.
12 00:00:53,071 –> 00:00:57,202 However then we need that the real part of z is positive.
13 00:00:58,114 –> 00:01:02,640 Now this means it’s a allowed to put in for example natural numbers.
14 00:01:03,514 –> 00:01:07,713 and indeed there we find a nice property for Gamma of n.
15 00:01:08,529 –> 00:01:13,434 Because one can show this is exactly (n-1)!
16 00:01:13,871 –> 00:01:18,593 Or in other words the Gamma function is a generalisation for the factorial.
17 00:01:19,500 –> 00:01:24,884 and exactly this gets even more apparent when we prove a similar recursive formula.
18 00:01:25,729 –> 00:01:33,627 Which reads like Gamma of (z+1) is equal to z time Gamma of z.
19 00:01:34,786 –> 00:01:40,744 Of course the Gamma function has also other nice properties we can discuss in another video.
20 00:01:41,514 –> 00:01:45,300 However now you already know what the definition of the Gamma function is.
21 00:01:45,886 –> 00:01:49,260 So if this was helpful, then i see you next time.
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Quiz Content
Q1: What is the value of $\Gamma(1)$?
A1: $1$
A2: $0$
A3: $-1$
A4: $2$
Q2: Is $\Gamma(1+i)$ defined?
A1: Yes!
A2: No!
A3: One needs more information.
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Last update: 2024-11
