00:00 Intro

01:21 Definition of open set in ℂ

03:22 Definition of differentiability in ℂ

07:01 Endcard

1 00:00:00,571 –> 00:00:03,935 Hello and welcome back to complex analysis.

2 00:00:04,729 –> 00:00:11,186 and first i want to thank all the nice people that support this channel on Steady, via Paypal or by other means.

3 00:00:12,029 –> 00:00:19,157 Now, in today’s part 2 we will talk about the complex differentiability of functions from C to C.

4 00:00:19,986 –> 00:00:26,007 This means that the input and the output of the function f is given by complex numbers.

5 00:00:26,957 –> 00:00:32,900 Now you might recall from real analysis, that first differentiability is a local property.

6 00:00:33,929 –> 00:00:39,871 This means when we ask the question, if the function f is differentiable at a given point z_0

7 00:00:39,986 –> 00:00:43,608 it only matters what happens around this point z_0.

8 00:00:44,186 –> 00:00:50,013 and for that reason the domain of the function f does not have to be the whole complex plane.

9 00:00:50,586 –> 00:00:54,622 It can be any open set U in the complex plane.

10 00:00:55,386 –> 00:01:01,100 Hence if you visualize the complex plane like this, you can just mark any region here.

11 00:01:01,829 –> 00:01:06,938 For example our domain U for the function f could be this set here.

12 00:01:07,614 –> 00:01:13,350 It could look as strange as you want. The only important thing we need is that it’s an open set.

13 00:01:14,271 –> 00:01:20,095 Now of course the natural question now should be: what is an open set for the complex numbers.

14 00:01:21,229 –> 00:01:27,514 Hence this is the first thing we now want to define before we talk about the definition of differentiability.

15 00:01:28,343 –> 00:01:32,921 Now I can tell you this definition is the same for all metric spaces.

16 00:01:33,121 –> 00:01:37,741 and please recall a metric space is a set, where you can measure distances.

17 00:01:38,571 –> 00:01:45,183 In C, the complex numbers, we can do this and of course we can also do this in R, the real numbers.

18 00:01:45,771 –> 00:01:51,324 Therefore this definition is exactly the same as I showed you in my real analysis course.

19 00:01:52,029 –> 00:01:58,160 Nevertheless it’s good to refresh your memory because this notion is used throughout this course.

20 00:01:59,129 –> 00:02:02,333 Indeed openness is not complicated at all.

21 00:02:02,486 –> 00:02:06,222 and to get an idea let’s take our set U from before.

22 00:02:06,771 –> 00:02:15,680 Visually speaking this set should be an open set if these boundary points you can see here are not part of the set U itself.

23 00:02:16,343 –> 00:02:25,245 In other words if you are a point in the set, then around you, you only see points of the set U.

24 00:02:26,014 –> 00:02:31,913 To put it more precisely; a whole epsilon-ball around z lies in the set U.

25 00:02:32,571 –> 00:02:37,505 and if this works for every point from the set U, we call U open.

26 00:02:38,257 –> 00:02:42,796 So formally we would write: for all z in U,

27 00:02:43,471 –> 00:02:46,324 there exists an epsilon-ball

28 00:02:46,929 –> 00:02:52,956 and here you already know from the last video, the epsilon-ball is denoted by B_epsilon(z).

29 00:02:53,929 –> 00:02:59,529 and now our claim here is, this epsilon-ball lies completely inside the set U.

30 00:03:00,500 –> 00:03:05,244 Ok, so this is openness for subsets of the complex numbers.

31 00:03:05,900 –> 00:03:10,124 One important example is of course the complex plane itself.

32 00:03:10,943 –> 00:03:16,627 Also it’s easy to show, that every epsilon-ball is open with this definition.

33 00:03:17,614 –> 00:03:22,236 Ok, other useful and important examples we will see later.

34 00:03:22,971 –> 00:03:27,416 Here first as promised we want to talk about differentiability.

35 00:03:28,400 –> 00:03:33,003 However now the function f we consider has a domain we call U.

36 00:03:33,829 –> 00:03:38,679 So it’s still a complex function, but now with a general open domain U.

37 00:03:39,914 –> 00:03:46,886 and now to write down the definition of differentiable at a given point, we have to fix this given point.

38 00:03:47,571 –> 00:03:51,211 So we have z_0 as a point in U.

39 00:03:51,957 –> 00:03:58,658 Hence, then we are able to define what it means that the function f is differentiable at z_0.

40 00:03:59,529 –> 00:04:07,398 and often to make it clear that we are talking about complex functions here, we say that the function is complex differentiable at the point.

41 00:04:08,386 –> 00:04:12,643 However the meaning is exactly the same as for real functions.

42 00:04:13,643 –> 00:04:17,191 We just want a linear approximation around this point.

43 00:04:17,391 –> 00:04:20,958 Which means the slope at this point should exists.

44 00:04:21,586 –> 00:04:26,029 and of course a slope we usually calculate with a difference quotient.

45 00:04:27,014 –> 00:04:32,214 So we have the difference in the output divided by the difference in the input.

46 00:04:33,014 –> 00:04:37,248 In other words what you see here is the slope of a secant.

47 00:04:38,029 –> 00:04:42,436 and in order to get the slope of the tangent, we have to consider the limit.

48 00:04:42,871 –> 00:04:47,308 Hence, z goes to z_0 and this limit should exist.

49 00:04:47,886 –> 00:04:54,094 and in fact this is the whole definition of complex differentiability at a given point z_0.

50 00:04:54,857 –> 00:04:58,971 However I think I should say a few words what this limit means here.

51 00:04:59,800 –> 00:05:04,300 Of course as often the formal definition is the same as for real numbers.

52 00:05:05,043 –> 00:05:10,136 However, we immediately find different consequences when we deal with complex numbers.

53 00:05:11,000 –> 00:05:14,594 Before we discuss them let’s first look at the definition here.

54 00:05:15,543 –> 00:05:23,147 The definition of the limit means that we consider all sequences z_n in the set U, but they shouldn’t take the value z_0.

55 00:05:23,829 –> 00:05:27,684 However they all should converge to the point z_0.

56 00:05:28,343 –> 00:05:33,251 Now, for each of the sequences z_n, we get a new sequence here.

57 00:05:33,771 –> 00:05:39,100 and of course the claim here is that this new sequence here also converges.

58 00:05:39,417 –> 00:05:45,326 and also the limit we get out here does not depend on the chosen sequence z_n.

59 00:05:45,526 –> 00:05:47,909 We always get the same number out.

60 00:05:48,957 –> 00:05:53,275 So please remember this is just the meaning of this symbol here.

61 00:05:53,886 –> 00:05:58,174 Indeed in real analysis we have exactly the same definition.

62 00:05:58,843 –> 00:06:02,526 However there you can split the limit into two parts.

63 00:06:03,286 –> 00:06:07,749 You could approximate this from the right hand side or the left hand side,

64 00:06:08,343 –> 00:06:12,351 because in R, in the real numbers, we have an order.

65 00:06:12,551 –> 00:06:16,168 and you already know we don’t have an order in C.

66 00:06:17,043 –> 00:06:22,594 Indeed, if you want to converge into this point z_0, we have a lot of directions.

67 00:06:23,343 –> 00:06:33,822 So you immediately see, with all these directions this limit here is more complicated than the real differentiability, we have in real analysis.

68 00:06:34,729 –> 00:06:39,771 So it looks the same, but here we immediately have a 2 dimensional problem.

69 00:06:40,757 –> 00:06:47,569 Ok, I would say, how all this is related to the linear approximation as we have it for real functions,

70 00:06:47,769 –> 00:06:49,843 we will discuss in the next video.

71 00:06:50,243 –> 00:06:56,711 and then we also look at examples and then you will see why complex analysis is so interesting.

72 00:06:57,571 –> 00:07:00,059 Therefore I hope I see you there and have a nice day.

73 00:07:00,259 –> 00:07:01,100 Bye!

Q1: What is the correct definition for a subset $U \subseteq \mathbb{C}$ being open?

A1: For all $z \in U$, there is another open set $V$ with $z \in V$.

A2: For all $z \in U$, there is an $\varepsilon$-ball $B_{\varepsilon}(z)$ with $z \in B_{\varepsilon}(z)$.

A3: For all $z \in U$, there is an $\varepsilon$-ball $B_{\varepsilon}(z)$ with $B_{\varepsilon}(z) \subseteq U$.

A4: For all $z \in U$, there is an $\varepsilon$-ball $B_{\varepsilon}(z)$ with $B_{\varepsilon}(z) \supseteq U$.

Q2: Which of the following subsets of $\mathbb{C}$ is open?

A1: $\mathbb{R}$

A2: $[-1,1]$

A3: ${ z \in \mathbb{C} \mid \mathrm{Re}(z) \geq 0 }$

A4: ${ z \in \mathbb{C} \mid \mathrm{Re}(z) = 5 }$

A5: $\mathbb{C}$

Q3: What is not a correct definition for a function $f: \mathbb{C} \rightarrow \mathbb{C}$ being (complex) differentiable at $z_0 \in \mathbb{C}$?

A1: $\displaystyle \lim_{z \rightarrow z_0} \frac{f(z) - f(z_0)}{z - z_0}$ exists.

A2: $\displaystyle \lim_{z \rightarrow z_0} \frac{f(z_0) - f(z)}{z_0 - z}$ exists.

A3: $\displaystyle \lim_{x \rightarrow z_0} \frac{f(z_0) - f(x)}{z_0 - x}$ exists.

A4: $\displaystyle \lim_{z_0 \rightarrow z} \frac{f(z_0) - f(z_0)}{z - z_0}$ exists.

A5: $\displaystyle \lim_{n \rightarrow \infty} \frac{f(z_n) - f(z_0)}{z_n - z_0}$ if we get the same value for each sequence $z_n \xrightarrow{n \rightarrow \infty} z_0$.