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Title: Heaviside Function
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Series: Advent of Mathematical Symbols
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YouTube-Title: Heaviside Function
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Bright video: https://youtu.be/fstRjd2xBc8
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Dark video: https://youtu.be/eshN4oCYz7U
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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: aoms22_sub_eng.srt
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Timestamps (n/a)
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Subtitle in English
1 00:00:00,629 –> 00:00:02,135 Hello and welcome.
2 00:00:02,335 –> 00:00:06,129 and the mathematical symbol of today is the so called Heaviside function.
3 00:00:06,200 –> 00:00:08,302 Often denoted with a capital H.
4 00:00:09,214 –> 00:00:12,786 However you also often see the greek letter Theta.
5 00:00:13,686 –> 00:00:18,443 Moreover sometimes the Heaviside function is called the unit step function.
6 00:00:19,429 –> 00:00:23,783 This is because the definition of H(x) is very simple.
7 00:00:24,814 –> 00:00:26,507 There is just one step involved.
8 00:00:26,707 –> 00:00:28,319 So we can consider 2 cases.
9 00:00:29,143 –> 00:00:32,441 So either the function is 1 or 0.
10 00:00:33,400 –> 00:00:37,586 and indeed the sign of x tells us in which case we are.
11 00:00:38,586 –> 00:00:43,078 Positive points get the value 1 and negative points get the value 0.
12 00:00:44,014 –> 00:00:47,883 Therefore only the question remains what to do with the point 0.
13 00:00:48,686 –> 00:00:51,429 and as you can see here we put it to 1.
14 00:00:51,888 –> 00:00:55,129 However there are also other conventions what to do.
15 00:00:55,232 –> 00:00:58,722 Sometimes x = 0 is set to 0.5.
16 00:00:59,714 –> 00:01:02,425 Then the value lies exactly in the middle here.
17 00:01:03,286 –> 00:01:10,214 However actually for a lot of applications it does not matter at all what we do with the point x = 0.
18 00:01:11,114 –> 00:01:14,244 The important thing is just that we have this step here.
19 00:01:15,343 –> 00:01:21,153 Hence the graph of the function looks very simple. We just have these 2 constant parts.
20 00:01:21,929 –> 00:01:24,525 Ok now you know the Heaviside function.
21 00:01:24,743 –> 00:01:29,862 Which is by the way not called Heaviside, because there is a heavy side in the graph.
22 00:01:30,786 –> 00:01:35,289 It’s called Heaviside as in the mathematician Oliver Heaviside.
23 00:01:36,214 –> 00:01:40,558 Ok then just i tell you one last fun fact for the end of the video.
24 00:01:41,671 –> 00:01:45,943 You can calculate the derivative of H and you get something nice out.
25 00:01:46,743 –> 00:01:50,343 Of course what comes out should be 0 here and here
26 00:01:50,618 –> 00:01:53,457 and maybe not defined here at 0.
27 00:01:54,400 –> 00:01:57,880 However this only happens if you do a classical derivative.
28 00:01:58,571 –> 00:02:02,229 What one can do is use the derivative for distributions.
29 00:02:02,943 –> 00:02:07,386 and then what comes out is the famous Dirac delta distribution.
30 00:02:08,271 –> 00:02:12,987 This is a really nice connection if you see both functions as distributions.
31 00:02:13,671 –> 00:02:18,386 In fact if you want to know more about distributions, i have a whole series about them.
32 00:02:19,543 –> 00:02:21,200 Therefore i hope i see you there.
33 00:02:21,729 –> 00:02:23,500 Have a nice day. Bye!
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Quiz Content
Q1: Does the Heaviside function have a heavy side?
A1: Yes, but it is still named after the mathematician Oliver Heaviside.
A2: No, not really.
Q2: Is the Heaviside function $H: \mathbb{R} \rightarrow \mathbb{R}$ continuously differentiable?
A1: No, the derivative does not exist at $x = 0$.
A2: Yes, the derivative is the constant function.
A3: Yes, the derivative is the continuous delta function.
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Last update: 2024-11
