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Title: Introduction
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Series: Complex Analysis
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YouTube-Title: Complex Analysis 1 | Introduction
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Bright video: https://youtu.be/dEu5ie25U0Y
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Dark video: https://youtu.be/g6LmapJ6k70
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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: ca01_sub_eng.srt
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Timestamps
00:00 Introduction
02:13 What we need
03:18 Metric space
04:43 Sequences and convergence in ℂ
07:32 Continuity for complex functions
09:30 Endcard
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Subtitle in English
1 00:00:00,400 –> 00:00:05,029 Hello and welcome to a new series about complex analysis
2 00:00:06,000 –> 00:00:11,855 and before we start I really want to thank all the nice people that support this channel on Steady or Paypal.
3 00:00:12,571 –> 00:00:19,058 Now this is part 1 in this series and we just start with a short introduction and some basic definitions.
4 00:00:20,029 –> 00:00:26,510 So the topic is called complex analysis, because it deals with functions defined on the complex plane.
5 00:00:27,243 –> 00:00:33,279 More concretely we consider differentiable functions f with domain C and codomain C.
6 00:00:34,471 –> 00:00:42,178 So you see it’s different from our real analysis course, where we have the real number line R as domain and codomain.
7 00:00:43,057 –> 00:00:49,322 Now soon we will see that this makes a major difference in the definition of differentiability
8 00:00:50,114 –> 00:00:55,972 and from that a lot of strong theorems will follow. Which indeed don’t hold in real analysis.
9 00:00:57,114 –> 00:01:07,971 However since the real number line R is a subset of the complex plane C, we can often apply these theorems even if we just consider some real functions.
10 00:01:08,943 –> 00:01:17,128 For example a real problem, where complex analysis can be really helpful is calculating an improper Riemann integral.
11 00:01:18,371 –> 00:01:26,994 This could be the integral from - infinity to infinity of the function: (x * sin(x)) / (1 + x^2).
12 00:01:27,971 –> 00:01:33,442 Now indeed finding an antiderivative of this function is really, really hard.
13 00:01:34,471 –> 00:01:40,823 However just calculating the whole integral with complex analysis is surprisingly simple.
14 00:01:41,671 –> 00:01:49,196 Indeed the result we find is a number slightly larger than 1. Namely it’s pi divided by e.
15 00:01:50,143 –> 00:01:54,382 How to do this simple calculation here, we will learn in this course.
16 00:01:55,129 –> 00:01:59,057 However of course first we have to start with the basics.
17 00:01:59,686 –> 00:02:05,981 and I already told you in contrast to real analysis we immediately start with differentiability.
18 00:02:06,700 –> 00:02:10,100 The definition first looks the same as in real analysis,
19 00:02:10,194 –> 00:02:13,039 but it’s way more powerful as we will see.
20 00:02:14,171 –> 00:02:19,229 Now in order to understand this new definition, you will need to know some basic facts.
21 00:02:20,386 –> 00:02:27,386 First you need to know how to work with sets and of course you also need to know how to calculate with complex numbers.
22 00:02:28,357 –> 00:02:34,067 If you have problems there, you find everything you need in my “start learning mathematics” series.
23 00:02:34,771 –> 00:02:39,764 This is the whole playlist that covers all the basics you need for this course here.
24 00:02:40,414 –> 00:02:46,771 In addition, it’s also good if you already have a basic knowledge of some topics of real analysis.
25 00:02:47,471 –> 00:02:53,793 For example you should know what a continuous or a differentiable function from R to R is.
26 00:02:54,400 –> 00:02:59,629 Also could be very helpful for you, if you already know what a power series is.
27 00:03:00,800 –> 00:03:05,455 For all these topics of course my real analysis series can help you.
28 00:03:06,214 –> 00:03:11,893 However you don’t need to watch the whole course. Some videos about these topics might be sufficient.
29 00:03:13,243 –> 00:03:17,500 Ok, by knowing this I now think we can start with the course.
30 00:03:18,343 –> 00:03:23,800 The first part will be about some definitions we will definitely need throughout the course.
31 00:03:24,971 –> 00:03:30,316 First I can tell you, the complex numbers form a set with the distant function.
32 00:03:31,114 –> 00:03:34,694 Formally we call such a construction a metric space,
33 00:03:34,894 –> 00:03:37,793 but don’t worry it’s not complicated at all.
34 00:03:38,643 –> 00:03:43,087 It just means that the distance between two elements of the set makes sense.
35 00:03:44,200 –> 00:03:48,103 and of course we can immediately visualize this in the complex plane.
36 00:03:49,014 –> 00:03:53,556 This means that a complex number z can be found in this plane.
37 00:03:54,214 –> 00:03:59,743 and on the x-axis we find the real part of z and on the y-axis the imaginary part.
38 00:04:00,643 –> 00:04:05,478 Now you can imagine that we have a second complex number here, we call w.
39 00:04:06,657 –> 00:04:11,296 and now what we want to do is to measure the distance between both points.
40 00:04:12,400 –> 00:04:16,311 Indeed this is what we calculate with the absolute value in C.
41 00:04:17,671 –> 00:04:22,833 and what we need is the absolute value of the complex number (z - w).
42 00:04:24,029 –> 00:04:32,977 Now such a notion of a distance is important, because with a distance we can say: what a convergent sequence is, what limits are and so on.
43 00:04:33,814 –> 00:04:39,220 Otherwise we wouldn’t be able to say that a sequence gets closer and closer to a given point.
44 00:04:40,043 –> 00:04:42,761 Indeed we have to measure this closeness.
45 00:04:43,800 –> 00:04:49,464 Hence, now we are able to talk about sequences of complex numbers and convergent sequences.
46 00:04:50,429 –> 00:04:57,253 So a sequence of complex numbers is denoted by z_n, where n goes through all natural numbers.
47 00:04:58,071 –> 00:05:04,143 So formally we just have a map that goes from the natural numbers into the complex numbers C.
48 00:05:04,535 –> 00:05:08,160 However we will always write the sequence in this form.
49 00:05:08,929 –> 00:05:16,339 and now we are able to define what it means that this sequence is convergent to a fixed complex number “a”.
50 00:05:17,314 –> 00:05:26,035 It just means that the distance we can measure between z_n and “a” gets smaller and smaller until it reaches 0 in the limit.
51 00:05:26,886 –> 00:05:30,829 This means that we can just look at a sequence of real numbers.
52 00:05:31,486 –> 00:05:36,114 Namely the sequence of the absolute value of (z_n - a).
53 00:05:37,171 –> 00:05:41,092 This is but a definition of the absolute value, a sequence of real numbers.
54 00:05:41,292 –> 00:05:43,414 Of non-negative real numbers.
55 00:05:44,543 –> 00:05:49,343 and now in the case that this sequence of real numbers goes to the limit 0,
56 00:05:49,443 –> 00:05:54,157 we call the sequence of complex numbers z_n convergent to “a”.
57 00:05:54,829 –> 00:06:00,947 Now if you don’t remember the definition of convergence for real numbers, let’s recall it.
58 00:06:01,600 –> 00:06:10,768 So we find that for all positive numbers we can call epsilon, there exists an index, we can call capital N
59 00:06:11,957 –> 00:06:18,017 such that for all indices afterwards, so n is greater or equal than capital N,
60 00:06:18,943 –> 00:06:24,916 we get that the distance between z_n and the point “a” is less than epsilon.
61 00:06:25,614 –> 00:06:30,442 So you see this is exactly the definition we learned in real analysis.
62 00:06:31,400 –> 00:06:35,380 However now we can visualize it in the complex plane.
63 00:06:36,229 –> 00:06:44,086 Now, this formula with the distance less than epsilon means that we can draw a circle around “a” with radius epsilon.
64 00:06:44,957 –> 00:06:51,167 With this we get that eventually the sequence members z_n lie inside this circle.
65 00:06:52,257 –> 00:06:55,302 So only finitely many can lie outside.
66 00:06:56,257 –> 00:07:02,876 and because this whole picture here is so important, we call the inside of the circle an epsilon ball.
67 00:07:03,757 –> 00:07:08,162 and for the notation we use a capital B with index epsilon.
68 00:07:09,286 –> 00:07:14,055 Moreover the middle point “a” here, we put into parentheses afterwards.
69 00:07:14,671 –> 00:07:20,314 Now by definition this is the set of all the complex numbers w, with the property:
70 00:07:20,750 –> 00:07:25,555 |w - a| is less than epsilon.
71 00:07:26,614 –> 00:07:31,098 So this is the definition of an epsilon-ball in the complex plane.
72 00:07:31,829 –> 00:07:36,813 Ok, because you now know what a convergent sequence in the complex numbers is,
73 00:07:37,186 –> 00:07:42,296 you also know what a continuous function between C and C is.
74 00:07:43,000 –> 00:07:47,200 In fact it has exactly the same meaning as for real functions.
75 00:07:48,229 –> 00:07:55,722 Hence small deviations, small errors in the input should be translated into small errors in the output.
76 00:07:56,457 –> 00:07:59,734 Indeed this can be formulated with sequences.
77 00:08:00,829 –> 00:08:06,643 Therefore we say a function is continuous at the point z_0 in C
78 00:08:07,243 –> 00:08:12,400 if for all sequences of complex numbers called z_n, we have
79 00:08:12,930 –> 00:08:18,314 that if z_n is convergent to the point z_0, then also
80 00:08:18,986 –> 00:08:24,951 the images f(z_n) are convergent with limit f(z_0)
81 00:08:25,543 –> 00:08:31,834 Of course here we use a common notation that tells us that a sequence is convergent to a given point.
82 00:08:32,629 –> 00:08:43,525 No matter which notation we use, you should remember continuous at a given point just means that convergence in the input implies convergent in the output.
83 00:08:44,486 –> 00:08:49,446 Ok, so with this we have the definition of continuity for complex functions.
84 00:08:50,500 –> 00:08:55,907 Therefore the next thing we need to define is the important notion of differentiability.
85 00:08:56,614 –> 00:09:01,879 This is what we will do in the next video, where we actually start with complex analysis.
86 00:09:02,214 –> 00:09:08,971 Now if you have any problems with these definitions in this video here, please check out my real analysis course.
87 00:09:09,771 –> 00:09:13,414 There I talk a lot about these definitions for real functions.
88 00:09:14,214 –> 00:09:19,009 However if you understand it, you can immediately translate this to complex functions.
89 00:09:19,986 –> 00:09:25,373 Ok, please also note the pdf version of this video, you can find in the description.
90 00:09:25,573 –> 00:09:27,875 as well as a quiz about this topic.
91 00:09:28,486 –> 00:09:30,257 Then I hope I see you next time.
92 00:09:30,369 –> 00:09:31,243 Bye!
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Quiz Content
Q1: How to measure the distance between two complex numbers $z,w \in \mathbb{C}$
A1: $z-w$
A2: $|z+w|$
A3: $|z-w|$
A4: $z+w$
Q2: Let $ (z_n){n \in \mathbb{N}} $ be a sequence of complex numbers. What is the correct definition of ‘The sequence $(z_n){n \in \mathbb{N}}$ is convergent to $a$’.
A1: $\forall \varepsilon > 0 ~~ \exists N \in \mathbb{N} ~~ \forall n \leq N ~:~ |z_n - a| < \varepsilon$.
A2: $\forall \varepsilon > 0 ~~ \exists N \in \mathbb{N} ~~ \forall n \geq N ~:~ |z_n - a| < \varepsilon$.
A3: $\exists \varepsilon > 0 ~~ \exists N \in \mathbb{N} ~~ \forall n \leq N ~:~ |z_n - a| < \varepsilon$.
A4: $\exists \varepsilon > 0 ~~ \forall N \in \mathbb{N} ~~ \forall n \leq N ~:~ |z_n - a| < \varepsilon$.
Q3: Which of the following sequences is convergent?
A1: $( i^n )_{n \in \mathbb{N}}$
A2: $( (-2)^n )_{n \in \mathbb{N}}$
A3: $( i^n \frac{1}{n} )_{n \in \mathbb{N}}$
A4: $( 2^n )_{n \in \mathbb{N}}$
Q4: What is the correct definition of the $\varepsilon$-ball with centre $a$?
A1: $B_{\varepsilon}(a) = { z \in \mathbb{C} \mid z = 1 }$
A2: $B_{\varepsilon}(a) = { z \in \mathbb{C} \mid z - a = \varepsilon }$
A3: $B_{\varepsilon}(a) = { z \in \mathbb{C} \mid |z-a| = \varepsilon }$
A4: $B_{\varepsilon}(a) = { z \in \mathbb{C} \mid |z-a| < \varepsilon }$
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Last update: 2024-10
