00:00 Intro

00:57 Why are power series important? Example of exp(z)

02:05 General definition

04:03 Example. Geometric series + conditions for convergence

09:06 Cauchy-Hadamard theorem

1 00:00:00,743 –> 00:00:04,213 Hello and welcome back to complex analysis.

2 00:00:04,914 –> 00:00:11,503 and first many, many thanks to all the nice people that support this channel on Steady, via Paypal or by other means.

3 00:00:12,214 –> 00:00:17,441 Please don’t forget, as a supporter you get a PDF version and a quiz for this video.

4 00:00:18,300 –> 00:00:21,671 Now, todays part 9 will be about power series.

5 00:00:22,471 –> 00:00:26,872 Indeed, this is a very important topic in a complex analysis course.

6 00:00:27,586 –> 00:00:33,385 In the case you have watched my real analysis course, you already know a lot about power series.

7 00:00:34,543 –> 00:00:41,147 Now it turns out, for this concept in the complex numbers, we can do exactly the same as in the real numbers.

8 00:00:41,771 –> 00:00:48,785 Moreover by working with complex numbers we will find the true nature of the behavior of some power series.

9 00:00:49,829 –> 00:00:55,419 Hence you need to work in the complex realm to understand power series completely.

10 00:00:55,871 –> 00:01:00,172 and why this is important, I can immediately show you with an example.

11 00:01:00,700 –> 00:01:05,916 Namely one of the most important functions is given by a power series.

12 00:01:07,043 –> 00:01:11,196 This is the exponential function denoted by exp(z).

13 00:01:11,743 –> 00:01:16,634 and now you know the value of the function is defined by an infinite sum.

14 00:01:16,834 –> 00:01:21,188 There we have z^k divided by k!.

15 00:01:22,286 –> 00:01:25,832 Now, this definition you should know for real numbers,

16 00:01:26,486 –> 00:01:32,483 but then we don’t have any problems also putting complex numbers into the formula.

17 00:01:32,857 –> 00:01:37,037 Of course then in general the outcome is also a complex number.

18 00:01:37,886 –> 00:01:43,114 Here please don’t forget, this symbol infinity means there is a limit involved.

19 00:01:43,843 –> 00:01:52,949 This means we have a limit of complex numbers and because you know how to measure distances in a complex plane, you don’t have any problem with this.

20 00:01:53,857 –> 00:01:58,254 Regarding this there is simply not much difference to the real counterpart.

21 00:01:59,414 –> 00:02:06,387 Now with this philosophy we can also talk about the general definition of a power series in the complex field.

22 00:02:06,900 –> 00:02:12,953 In other words if you know the definition for real numbers, you know exactly what comes now.

23 00:02:13,300 –> 00:02:20,572 Of course the starting point should be the coefficients we have in the infinite series and are now complex numbers.

24 00:02:21,257 –> 00:02:25,255 and a fitting name is just a_0, a_1, a_2 and so on.

25 00:02:25,686 –> 00:02:31,350 Ok, then for these numbers we can define a complex function, we can call f.

26 00:02:31,986 –> 00:02:37,129 Moreover the domain we call D. So we have a function from D into C.

27 00:02:38,257 –> 00:02:48,919 Then this map is defined by sending z to the infinite series of the coefficient a_k times z^k.

28 00:02:49,543 –> 00:02:58,530 However, since later it will be important that we can shift the whole function by a fixed number z_0, I already introduce it here.

29 00:02:59,171 –> 00:03:02,671 So we simply have (z - z_0)^k.

30 00:03:03,257 –> 00:03:08,469 and I can already tell you this z_0 is often called the expansion point.

31 00:03:09,600 –> 00:03:15,134 Of course what is still missing here is the explicit definition of the domain D.

32 00:03:15,671 –> 00:03:21,755 However you might already know this, because here in the definition of the map there is a limit involved.

33 00:03:22,643 –> 00:03:28,183 And maybe this limit does not exist for all complex numbers z we can choose.

34 00:03:28,771 –> 00:03:34,135 Hence this restriction is now what we have to put into the definition of D.

35 00:03:34,600 –> 00:03:39,652 So we can say: we have all the complex numbers z, where this limit exists.

36 00:03:40,386 –> 00:03:44,101 Or in other words we simply say the series is convergent.

37 00:03:45,314 –> 00:03:54,198 Ok, then you see we have a well-defined function. Maybe the domain is very small, but still we always call this function a power series.

38 00:03:55,186 –> 00:03:58,585 With this knowledge I would say we look at an example.

39 00:03:59,600 –> 00:04:03,661 This is an important one and you already know it from real analysis.

40 00:04:04,457 –> 00:04:07,066 It’s the famous geometric series.

41 00:04:08,029 –> 00:04:12,603 It’s not complicated at all, because all coefficients are just 1.

42 00:04:13,014 –> 00:04:18,190 Therefore we have the series that start with k=0 of z^k.

43 00:04:19,357 –> 00:04:29,450 Now, in fact by using exactly the same proof as in real analysis, we can show that this series has the value 1/(1-z).

44 00:04:30,214 –> 00:04:36,917 However you might already know, this proof only works if the absolute value of z is less than 1.

45 00:04:37,600 –> 00:04:43,629 Indeed if the absolute value of z is greater or equal than 1, then this series does not converge.

46 00:04:44,543 –> 00:04:50,695 In other words the domain D here is the ball around 0 with radius 1.

47 00:04:51,457 –> 00:04:58,070 and this is such an important fact, that you really should remember it, because we can use it a lot in the future.

48 00:04:58,643 –> 00:05:02,614 Moreover please recall the notation we use for the epsilon ball.

49 00:05:02,657 –> 00:05:09,871 So it’s the open ball, so we call it B and the radius is 1. So we have an index 1 and the middle point is just 0.

50 00:05:11,057 –> 00:05:21,230 Ok, so what you see here in the complex plane: we have a circle and inside we have convergence and outside we have divergence.

51 00:05:22,300 –> 00:05:29,387 Now it turns out that for any power series this situation is not essentially more difficult.

52 00:05:29,729 –> 00:05:33,905 Indeed the following fact follows from the geometric series.

53 00:05:34,629 –> 00:05:41,355 It simply tells us that for a given power series, there is a maximal radius r for this circle here.

54 00:05:41,986 –> 00:05:48,201 Or to say it more concretely, the open ball with radius r lies completely in D.

55 00:05:49,014 –> 00:05:56,446 So you see the only difference from before is that we don’t need equality and also the middle point should be z_0 now.

56 00:05:56,957 –> 00:06:04,786 So here you see the worst case for r would be to be 0, which means there is only 1 point in D, z_0.

57 00:06:05,600 –> 00:06:09,464 and the best-case scenario would be that r is infinity.

58 00:06:09,529 –> 00:06:12,476 Which means D is just the whole complex plane.

59 00:06:12,986 –> 00:06:24,310 So by using formulas we would write: r comes from the interval 0 to infinity. However, also infinity should be included as a symbol.

60 00:06:24,857 –> 00:06:29,238 Therefore to be precise we would distinguish between 2 cases here.

61 00:06:30,343 –> 00:06:35,232 Of course I told you before, the second case is just that D is the whole complex plane.

62 00:06:35,786 –> 00:06:40,545 This is the best case, because there the power series converges everywhere.

63 00:06:41,186 –> 00:06:49,149 In the other case please note that we know, that somewhere on the boundary or outside of the ball, we have divergence.

64 00:06:49,700 –> 00:06:57,705 Indeed, because this r is chosen maximally, the result is that everywhere outside we have divergence.

65 00:06:58,657 –> 00:07:01,798 Therefore the general picture looks like this.

66 00:07:02,571 –> 00:07:13,149 In the complex plane we have this ball with z_0 in the middle and with radius r and in the inside of this ball convergence is guaranteed.

67 00:07:13,929 –> 00:07:18,307 Moreover in the outside of this ball, divergence is guaranteed.

68 00:07:18,943 –> 00:07:23,040 The only thing we don’t know in general is what happens at the boundary.

69 00:07:23,986 –> 00:07:28,171 Indeed, depending on the power series different things could happen there.

70 00:07:29,171 –> 00:07:34,137 For example for the geometric series the boundary was completely divergent as well.

71 00:07:35,214 –> 00:07:40,328 However, for a different power series you could have convergence or both cases mixed.

72 00:07:40,886 –> 00:07:45,569 Ok, then let’s state the divergence part here also with a formula.

73 00:07:46,286 –> 00:07:52,104 Now, the outside in this complex plane here, could be described as this set difference.

74 00:07:52,657 –> 00:07:56,914 So it simply means that z is not an element of this ball.

75 00:07:57,314 –> 00:08:02,790 However, then the boundary would be included so we have to exclude this as well.

76 00:08:03,514 –> 00:08:07,582 We can simply do this by taking the closure of this ball.

77 00:08:08,157 –> 00:08:12,835 So this notation simply means: this is the set including the boundary.

78 00:08:13,543 –> 00:08:17,434 So in this case it’s not the open ball, it’s the closed ball.

79 00:08:18,329 –> 00:08:24,769 Now, for points that are not in this closed ball we know that the power series here is divergent.

80 00:08:25,529 –> 00:08:32,806 So in summary you should see this here is a very nice result, we can use a lot when we deal with power series.

81 00:08:33,543 –> 00:08:42,093 Indeed, it’s not hard to prove. We can just use the root criterion from real analysis and combine it with the geometric series here.

82 00:08:43,071 –> 00:08:48,112 Ok, if you are interested in seeing this proof, please let me know in the comments.

83 00:08:48,729 –> 00:08:54,714 Now, for the end here I can tell you, often it’s sufficient to know that there exists such an r,

84 00:08:54,829 –> 00:08:58,429 but sometimes it would be nice to have an explicit formula for it.

85 00:08:59,157 –> 00:09:02,674 Then we would be able to calculate this value of r.

86 00:09:03,371 –> 00:09:08,992 Probably you already know this formula. It’s often called the Cauchy-Hadamard theorem.

87 00:09:09,771 –> 00:09:15,386 In fact this is what we have discussed in my real analysis course in part 33.

88 00:09:16,071 –> 00:09:21,622 and it tells you that 1/r can be calculated by a lim sup.

89 00:09:22,457 –> 00:09:31,101 Namely we look at the coefficient a_k in the absolute value and then we take the k-th root of this real number.

90 00:09:32,114 –> 00:09:37,782 Then we take the lim sup k to infinity and we know this always exists.

91 00:09:38,329 –> 00:09:43,537 It’s either a real number between 0 and infinity or this symbol infinity.

92 00:09:44,543 –> 00:09:49,308 and then simply the inverse of this number gives us our maximal r.

93 00:09:49,757 –> 00:09:56,027 There of course you should know, inverses with 0 and infinity are here defined by this formula.

94 00:09:56,700 –> 00:10:01,844 Hence the best-case scenario is: in this lim sup is 0.

95 00:10:02,514 –> 00:10:11,727 Ok, the last important thing I should tell you is that this maximal r is often called the radius of convergence for the power series.

96 00:10:12,500 –> 00:10:19,729 and I would say it’s a good exercise for you to calculate the radius of convergence for the exponential function from above.

97 00:10:20,486 –> 00:10:25,984 Ok, then I hope I see you next time, when we start talking about the uniform convergence.

98 00:10:26,686 –> 00:10:28,827 So have a nice day and see you there.

99 00:10:28,828 –> 00:10:29,700 Bye!

Q1: For which $q \in \mathbb{C}$ is the series $\displaystyle \sum_{k = 0}^\infty q^k$ convergent?

A1: Only for $q = 0$

A2: For none.

A3: Only for $|q| < 1$.

A4: Only for $|q| \leq 1$.

A5: Only for $|q| \geq 1$.

Q2: Which statement is correct?

A1: For all $q \in \mathbb{C}$, we have $\displaystyle \sum_{k = 0}^\infty q^k = \frac{1}{1 - q}$

A2: For all $|q| < 1$, we have $\displaystyle \sum_{k = 0}^\infty q^k = \frac{1}{1 - q}$

A3: For all $|q| < 1$, we have $\displaystyle \sum_{k = 0}^\infty q^k = \frac{1}{1 + q}$

A4: For all $|q| > 1$, we have $\displaystyle \sum_{k = 0}^\infty q^k = \frac{2}{1 + q}$

Q3: What is the correct definition of the exponential function $\exp$.

A1: $\exp(z) := \displaystyle \sum_{k = 1}^\infty \frac{z^k}{k!}$

A2: $\exp(z) := \displaystyle \sum_{k = 0}^\infty \frac{z^k}{k!}$

A3: $\exp(z) := \displaystyle \sum_{k = 0}^\infty \frac{z^k}{k}$

Q4: Which statement is true for each power series $\sum_{k=0}^\infty a_k z^k$ with domain $\mathcal{D}$?

A1: The point $z=0$ lies in $\mathcal{D}$.

A2: The domain is $\mathcal{D} = \mathbb{C}$.