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Title: Uniform Convergence
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Series: Complex Analysis
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YouTube-Title: Complex Analysis 10 | Uniform Convergence
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Bright video: https://youtu.be/qSBdxI3rI04
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Dark video: https://youtu.be/p8I0E-P8WY0
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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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Subtitle on GitHub: ca10_sub_eng.srt
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Timestamps
0:00 Intro
0:40 Uniform convergence definition
2:17 Power series are holomorphic
2:31 Correction: codomain for the power series should be given by ℂ
3:25 Uniform convergence for power series
7:01 Credits
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Subtitle in English
1 00:00:00,586 –> 00:00:03,843 Hello and welcome back to complex analysis.
2 00:00:04,714 –> 00:00:11,081 and before we start I want to thank all the nice people that support this channel on Steady, via Paypal or by other means.
3 00:00:12,057 –> 00:00:15,895 Now, here you see we’ve already reached part 10 in this series.
4 00:00:16,557 –> 00:00:22,443 Therefore it’s time to start talking about the important concept of uniform convergence.
5 00:00:22,829 –> 00:00:28,057 The definition for this convergence is indeed exactly the same as in real analysis.
6 00:00:28,786 –> 00:00:34,984 Moreover you will also see that in complex analysis we can immediately apply it to the power series.
7 00:00:35,614 –> 00:00:39,334 However let’s first start with the formal definition.
8 00:00:39,534 –> 00:00:46,134 So what we need is a sequence of functions. Now they are complex functions with the same domain U.
9 00:00:46,629 –> 00:00:50,668 This means for each natural number n we have a function f_n here.
10 00:00:51,643 –> 00:00:56,052 Additionally we need another function f, which should be our limit.
11 00:00:56,443 –> 00:01:03,178 Hence we say: the sequence of functions f_n is uniformly convergent to another function f,
12 00:01:03,771 –> 00:01:11,065 if the supremum norm of (f_n - f) goes to 0 when n goes to infinity.
13 00:01:11,614 –> 00:01:16,642 and as usual the supremum norm is denoted with an infinity symbol in the index.
14 00:01:17,429 –> 00:01:22,307 Now in the case that you have never seen the supremum norm, let’s quickly define it.
15 00:01:23,086 –> 00:01:28,721 In fact the only thing we need is that the values of the functions that are involved are numbers.
16 00:01:29,357 –> 00:01:35,743 Then it does not matter if we have real or complex numbers, because we can always take the absolute value.
17 00:01:36,571 –> 00:01:41,500 and then we are just interested in the largest value that can come out or more precisely
18 00:01:41,529 –> 00:01:44,829 we look at the supremum where we go through all numbers z.
19 00:01:45,857 –> 00:01:53,308 and now what we want for the uniform convergence is that this supremum goes to 0, when n increases to infinity.
20 00:01:54,200 –> 00:02:02,130 So please note, we don’t just fixed the point z and look at the convergence, we look at all points at the same time in the limit.
21 00:02:02,586 –> 00:02:06,640 and exactly this makes the convergence here uniform.
22 00:02:07,357 –> 00:02:12,170 Ok, now this notion helps us a lot when we deal with power series.
23 00:02:13,086 –> 00:02:18,107 In fact this is such an important result that we can use it a lot in complex analysis.
24 00:02:19,129 –> 00:02:24,160 Essentially it tells us that all power series are holomorphic functions.
25 00:02:24,829 –> 00:02:29,974 Later we will see that in some sense the reverse implication is also true.
26 00:02:30,586 –> 00:02:36,065 However let’s first start with the statement here, which holds in general for all power series.
27 00:02:36,686 –> 00:02:42,600 So you see what we have is a power series with coefficients a_k and expansion point z_0.
28 00:02:43,529 –> 00:02:48,583 Now, the only thing we put in is that the radius of convergence is greater than 0.
29 00:02:49,300 –> 00:02:52,165 It could be infinity, but it’s not 0.
30 00:02:52,614 –> 00:02:58,337 To say it differently the power series here is convergent at a different point than z_0.
31 00:02:58,971 –> 00:03:04,299 Therefore we can define this function on the whole open ball B_r(z_0).
32 00:03:04,957 –> 00:03:10,551 I always call it ball, but you now, in the 2-dimensional complex plane it’s a disk.
33 00:03:11,114 –> 00:03:15,137 Therefore sometimes I speak of the open disk around z_0.
34 00:03:15,971 –> 00:03:24,362 Now, from the last video you know inside this open disk we have convergence, but now we see, we also have uniform convergence.
35 00:03:25,371 –> 00:03:33,659 In fact we have this for this power series here, when we see it as a function defined on B_c(z_0).
36 00:03:34,414 –> 00:03:39,886 So we have a number c as a radius involved, which is smaller than the radius of convergence.
37 00:03:40,914 –> 00:03:47,854 and this is very important, because we don’t know if we have convergence on the boundary of this disk here.
38 00:03:48,486 –> 00:03:56,115 However, if we make our ball a little bit smaller, now we know we have also convergence on the boundary, on the circle.
39 00:03:57,343 –> 00:04:01,508 Therefore we can also include the boundary in this statement here.
40 00:04:02,329 –> 00:04:06,199 and there you know, the closed ball is denoted with this line here.
41 00:04:07,071 –> 00:04:14,674 Ok, if you are not used to working with series and power series, here you see an alternative formulation of this statement.
42 00:04:15,286 –> 00:04:22,228 I simply give you the explicit definition of sequence of functions f_n, we need for the uniform convergence.
43 00:04:22,871 –> 00:04:26,470 and you see, it’s defined by using partial sums.
44 00:04:27,329 –> 00:04:33,211 and now this sequence of functions converges uniformly to the function f.
45 00:04:33,411 –> 00:04:38,714 However, only when we restrict f_n to the closed ball B_c(z_0).
46 00:04:39,686 –> 00:04:47,772 Ok, so this is the first statement and the next 2 will tell us that the power series is indeed a holomorphic function.
47 00:04:48,643 –> 00:04:52,806 In order to show this we have to look at the formal complex derivative here.
48 00:04:53,757 –> 00:04:58,165 So the power k comes in front and we reduce the exponent by 1.
49 00:04:58,800 –> 00:05:06,514 We already know this works for polynomials. So when we have an end index here, but we don’t know if this also works for power series.
50 00:05:07,386 –> 00:05:11,404 But you will believe me, this is exactly what we want to prove now.
51 00:05:11,857 –> 00:05:19,541 The first step for this is that similarly to before we also get an uniform convergence for this new power series here.
52 00:05:20,300 –> 00:05:29,276 More precisely here we can also say, we have a sequence of functions f_n’ that converge uniformly to this function here.
53 00:05:29,814 –> 00:05:36,924 Indeed here the name f_n’ is justified, because it’s the complex derivative of this polynomial here.
54 00:05:37,929 –> 00:05:46,080 Of course in the end this is what we want when we have the derivative of f. We want to approximate it with these derivatives here.
55 00:05:46,714 –> 00:05:51,463 and obviously exactly this should be our last statement, three.
56 00:05:51,914 –> 00:05:58,856 Hence we can write: the function f is complex differentiable and the derivative is given by this formula.
57 00:05:59,471 –> 00:06:05,264 So maybe this is not such a big surprise, but still it’s a very nice result for us.
58 00:06:06,029 –> 00:06:12,843 and here please recall f is defined on an open domain and complex differentiable at all points.
59 00:06:13,043 –> 00:06:17,748 Hence, the power series gives us immediately a holomorphic function.
60 00:06:18,557 –> 00:06:23,544 So with this result we immediately get a lot of examples for holomorphic functions.
61 00:06:24,271 –> 00:06:31,181 In summary, in order to understand complex analysis, you really should remember this whole result here.
62 00:06:32,129 –> 00:06:40,602 In a few words we could say this result tells us that each power series is holomorphic on its open disk of convergence.
63 00:06:41,743 –> 00:06:46,201 Of course if you really to believe this result, we should prove it.
64 00:06:46,957 –> 00:06:50,777 However this will take some time. So let’s do this in the next video.
65 00:06:51,286 –> 00:06:55,143 Therefore I hope I see you there and have a nice day. Bye!
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Quiz Content
Q1: Let the power series $f: \mathbb{C} \rightarrow \mathbb{C}$ be given by $f(z) = \sum_{k=0}^\infty a_k z^k$. The radius of convergence is $\infty$. Which statement is not correct?
A1: For all $z \in \mathbb{C}$, the series $\sum_{k=0}^\infty a_k z^k$ is convergent.
A2: For all $z \in \mathbb{C}$, the series $\sum_{k=0}^\infty a_k z^k$ is absolutely convergent.
A3: The sequence of functions given by $g_n: \mathbb{C} \rightarrow \mathbb{C}$, $g_n(z) = \sum_{k=0}^n a_k z^k$ is uniformly convergent to $f$.
A4: For all $c \in \mathbb{C}$, the sequence of functions given by $g_n: \overline{B_c(0)} \rightarrow \mathbb{C}$, $g_n(x) = \sum_{k=0}^n a_k z^k$ is uniformly convergent to $f$ on the disc $\overline{B_n(0)}$.
Q2: Let the power series $f: \mathbb{C} \rightarrow \mathbb{C}$ be given by $f(z) = \sum_{k=0}^\infty a_k z^k$. The radius of convergence is $\infty$. Is $f$ complex differentiable?
A1: No, nowhere.
A2: Yes, but only for the point $z = 0$.
A3: Yes, everywhere.
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Last update: 2024-10
