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Title: Complex Differentiability
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Series: Complex Analysis
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YouTube-Title: Complex Analysis 2 | Complex Differentiability
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Bright video: https://youtu.be/Jk5hxibwW80
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Dark video: https://youtu.be/baWkq2y_7R4
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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: ca02_sub_eng.srt
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Timestamps
00:00 Intro
01:21 Definition of open set in ℂ
03:22 Definition of differentiability in ℂ
07:01 Endcard
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Subtitle in English
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!
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Quiz Content
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 $z_n \xrightarrow{n \rightarrow \infty} z_0$.
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Last update: 2024-10
