00:00 Intro

00:28 How to define a holomorphic function?

02:29 Essential properties of holomorphic functions

04:21 Example 1. A complex polynomial

06:11 Example 2. A ‘rational function’ for polynomials

1 00:00:00,629 –> 00:00:03,945 Hello and welcome back to complex analysis.

2 00:00:05,000 –> 00:00:11,565 and as always, first I want to thank all the nice people that support this channel on Steady, via Paypal or by other means.

3 00:00:12,457 –> 00:00:16,621 Now, this part 4 is about holomorphic and entire functions.

4 00:00:17,257 –> 00:00:24,467 So you see, we are still at the beginning of this interesting field such that we first have to fix the manner of speaking here.

5 00:00:25,343 –> 00:00:30,692 In fact the definition of a holomorphic function is what we need throughout this course.

6 00:00:31,586 –> 00:00:33,755 Therefore let’s immediately start with this.

7 00:00:33,955 –> 00:00:37,243 So we consider a function f with domain U.

8 00:00:38,057 –> 00:00:41,329 and of course this should be a complex valued function.

9 00:00:42,500 –> 00:00:48,715 Now, we have talked about this before. The domain U should always be an open subset of the complex numbers.

10 00:00:49,771 –> 00:00:55,967 So it does not matter how it looks like in the complex plane. The important thing is, it’s an open set.

11 00:00:56,786 –> 00:01:01,429 However later we will see that often we have a very nice looking domain.

12 00:01:02,657 –> 00:01:08,820 Nevertheless, still it’s very helpful to introduce holomorphic functions in this general sense.

13 00:01:09,743 –> 00:01:11,471 Hence let’s immediately do this.

14 00:01:11,829 –> 00:01:19,334 So a function f is called holomorphic, if f is complex differentiable at all points.

15 00:01:19,986 –> 00:01:23,571 So you see it’s not a complicated definition at all.

16 00:01:23,644 –> 00:01:29,386 It just tells us that the complex derivative makes sense no matter which point z_0 we choose.

17 00:01:30,429 –> 00:01:34,671 Therefore sometimes it can be helpful to emphasis the domain again.

18 00:01:34,943 –> 00:01:37,475 So we would say holomorphic on U.

19 00:01:38,357 –> 00:01:39,971 So what you should remember is

20 00:01:40,114 –> 00:01:47,900 a holomorphic function is just a complex valued function that is differentiable at all points of an open set.

21 00:01:48,414 –> 00:01:53,214 Now in the literature also see that other names than holomorphic are used.

22 00:01:54,243 –> 00:01:57,886 For example some people speak of regular functions.

23 00:01:58,600 –> 00:02:04,343 Moreover you also sometimes see the term complex analysis used for this definition here.

24 00:02:05,257 –> 00:02:09,608 However later i will tell you what this term should actually mean.

25 00:02:10,300 –> 00:02:12,930 It has something to do with power series.

26 00:02:13,857 –> 00:02:21,091 Ok, now there is another special name for holomorphic functions, whose domain is the whole complex plane.

27 00:02:22,157 –> 00:02:27,998 For this largest possible domain the holomorphic function is then called an entire function.

28 00:02:29,100 –> 00:02:33,743 So in other words, entire just means holomorphic on C.

29 00:02:34,357 –> 00:02:40,022 Ok, by knowing this we can now talk about some basic properties of holomorphic functions.

30 00:02:40,800 –> 00:02:45,818 Let’s start with the properties we can immediately adapt from the real valued counterparts.

31 00:02:47,014 –> 00:02:52,004 First we know the proof that differentiability implies continuity.

32 00:02:53,029 –> 00:02:59,392 Hence in complex analysis this means that a holomorphic function is always a continuous function.

33 00:03:00,257 –> 00:03:05,874 Indeed for showing this fact we can exactly redo the proof we did in real analysis.

34 00:03:06,829 –> 00:03:10,289 and this is what we can do for a lot of basic facts here.

35 00:03:11,014 –> 00:03:17,484 For example we still have our sum and product rule for derivatives, when we consider 2 functions.

36 00:03:18,457 –> 00:03:26,890 Therefore this implies, if f and g are holomorphic functions, the sum and the product are also holomorphic functions.

37 00:03:27,943 –> 00:03:32,335 Of course this is a fact, we will naturally use a lot in calculations.

38 00:03:33,571 –> 00:03:40,043 Ok, now besides the sum and the product rule, you also now the chain and quotient rule for derivatives.

39 00:03:40,857 –> 00:03:45,166 and also for them we can just redo the proof from real analysis.

40 00:03:46,171 –> 00:03:52,056 Therefore we can just state sum, product, quotient and chain rule still hold.

41 00:03:52,586 –> 00:03:58,784 For example you can check my video about the chain rule for real functions and when you see the proof,

42 00:03:58,786 –> 00:04:05,146 you will see that you can do the same steps, when you consider complex numbers instead of real numbers.

43 00:04:05,971 –> 00:04:12,429 Ok, by knowing this i think we are ready to talk about some important examples of holomorphic functions.

44 00:04:13,443 –> 00:04:19,755 Now, by simply using the rules from above we immediately see that polynomials are entire functions.

45 00:04:20,329 –> 00:04:26,586 More precisely this means that we have a function f, where the domain is the whole complex plane

46 00:04:26,976 –> 00:04:31,777 and where f(z) is given as a sum with fixed coefficient “a”.

47 00:04:32,486 –> 00:04:35,186 So we have a_m times z to the power m

48 00:04:35,243 –> 00:04:39,773

• a_(m-1) times z to the power (m-1)

49 00:04:39,973 –> 00:04:41,150

• and so on

50 00:04:41,200 –> 00:04:44,721 until we reach the constant term. The coefficient a_0.

51 00:04:45,371 –> 00:04:50,284 However here please keep in mind, the coefficients are fixed complex numbers.

52 00:04:51,443 –> 00:04:55,391 Now such a function here we call a complex polynomial.

53 00:04:55,591 –> 00:05:00,293 and if a_m is nonzero, we call m the degree of the polynomial.

54 00:05:01,043 –> 00:05:08,104 Of course here its not hard to see, using our sum and product rule, that a polynomial is a holomorphic function.

55 00:05:08,886 –> 00:05:13,595 and because there is no restriction in the domain, it’s also an entire function.

56 00:05:14,529 –> 00:05:21,129 Indeed just by using induction, you can calculate the derivative f’ for every point z.

57 00:05:21,729 –> 00:05:23,786 and now it might not surprise you

58 00:05:23,843 –> 00:05:29,329 this is m times a_m times z to the power (m-1)

59 00:05:30,171 –> 00:05:37,704

• (m-1) times a_(m-1) times z to the power (m-2)

60 00:05:38,357 –> 00:05:45,008

• and so on, until we reach 2 times a_2 times z to the power 1.

61 00:05:45,486 –> 00:05:49,387 and finally + the new constant term, a_1.

62 00:05:50,400 –> 00:05:53,625 So you see, this is what you really should remember.

63 00:05:53,825 –> 00:05:57,786 More precisely, when you want to calculate the derivative of such a term

64 00:05:57,829 –> 00:06:02,886 it just means you bring the exponent in front and reduce the exponent by 1.

65 00:06:03,857 –> 00:06:08,609 Hence the polynomial is still a nice function in the complex numbers.

66 00:06:09,400 –> 00:06:12,294 In fact the same holds for rational functions.

67 00:06:12,371 –> 00:06:15,129 Where we simply divide two polynomials

68 00:06:15,542 –> 00:06:21,353 However, because we have a denominator there, the domain can’t be the whole complex plane.

69 00:06:22,243 –> 00:06:26,052 This means that we have to exclude the zeros of the denominator.

70 00:06:26,957 –> 00:06:32,891 For example in the case that we consider the function f(z) is given by 1 over z.

71 00:06:33,586 –> 00:06:40,008 Then we are not allowed to divide by 0. Therefore we have to exclude 0 from the domain.

72 00:06:40,700 –> 00:06:47,231 Now the important thing to note here is, in the case we exclude finitely many points from the complex plane,

73 00:06:47,243 –> 00:06:49,242 we still get an open set.

74 00:06:50,214 –> 00:06:57,466 Therefore we can combine this fact with the quotient rule and conclude that this function is also holomorphic.

75 00:06:58,429 –> 00:07:03,730 Now, obviously this whole argumentation also works for any rational function.

76 00:07:04,800 –> 00:07:09,374 So in summary a rational function is also a holomorphic function.

77 00:07:10,157 –> 00:07:18,379 Now as a reminder the definition of a rational function is simply given as one polynomial p divided by another one, q.

78 00:07:19,000 –> 00:07:22,516 So f(z) = p(z) divided by q(z).

79 00:07:23,200 –> 00:07:29,374 Then the finite set S, we have to exclude here, is simply given by the zeros of q.

80 00:07:30,071 –> 00:07:34,141 Hence all points z in C, where q(z) = 0.

81 00:07:35,086 –> 00:07:38,548 Now such rational functions often occur in applications.

82 00:07:38,748 –> 00:07:42,116 Therefore it’s good to know that they are also holomorphic.

83 00:07:42,914 –> 00:07:47,357 However please keep in mind. In general they are not entire functions.

84 00:07:48,543 –> 00:07:55,654 Ok, with this we have some examples we can work with and of course later we will talk about other examples as well.

85 00:07:56,129 –> 00:08:00,814 But first, in the next video we will talk about the Cauchy–Riemann equations.

86 00:08:01,643 –> 00:08:06,977 In fact these equations will explain why complex analysis is so interesting.

87 00:08:07,571 –> 00:08:11,136 Therefore i really hope that i see you in the next video.

88 00:08:11,336 –> 00:08:12,171 Bye!

Q1: Which of the following functions $f: \mathbb{C} \rightarrow \mathbb{C}$ is not holomorphic?

A1: $f(z) = 1$

A2: $f(z) = z^2$

A3: $f(z) = z^3$.

A4: $f(z) = \overline{z}$

Q2: Can a function $f: \mathbb{C}\setminus { 0 } \rightarrow \mathbb{C}$ be an entire function?

A1: No, never!

A2: Yes, if it is holomorphic.

A3: No, because such a function can never be holomorphic.

A4: Yes, every such function is an entire function.

Q3: Which is not a property of the function $f: \mathbb{C}\setminus { 0 } \rightarrow \mathbb{C}$ given by $f(z) = \frac{1}{z}$.

A1: $f$ is a holomorphic function.

A2: $f(0)$ is not defined.

A3: $f$ is continuous.

A4: $f$ is an entire function.

A5: $f$ is complex differentiable at $z= i$.

A6: $f$ is a rational function.