• Title: Heaviside Function

• YouTube-Title: Advent of Mathematical Symbols - Part 22 - Heaviside Function

• Bright video: https://youtu.be/fstRjd2xBc8

• Dark video: https://youtu.be/eshN4oCYz7U

• Timestamps
• 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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