00:00 Intro

00:18 Quick recap

02:42 Example 1: Identity function

04:28 Example 2: Complex conjugate function

05:38 Example 3: Complex polynomial

1 00:00:00,457 –> 00:00:03,626 Hello and welcome back to complex analysis.

2 00:00:04,586 –> 00:00:10,827 and first I want to thank all the very nice people that support this channel on Steady, via Paypal or by other means.

3 00:00:11,743 –> 00:00:17,131 Now, in today’s part 7 I show you examples for the famous Cauchy-Riemann equations.

4 00:00:18,100 –> 00:00:22,444 For this let’s first recall the important theorem from the last video.

5 00:00:22,800 –> 00:00:28,554 There we introduced the Cauchy-Riemann equations with respect to the complex differentiability.

6 00:00:29,343 –> 00:00:33,884 Hence we can immediately reformulate the theorem for holomorphic functions.

7 00:00:34,700 –> 00:00:39,798 and as always for this we have to choose an open subset capital U in C.

8 00:00:40,400 –> 00:00:47,649 and now you should already know the complex function can be equivalently described as a function from R^2 into R^2.

9 00:00:48,200 –> 00:00:54,424 However in this case for the new function the domain should also be an open subset of R^2.

10 00:00:55,014 –> 00:00:57,625 So maybe let’s simply call it U_R.

11 00:00:58,686 –> 00:01:04,217 Obviously when you visualize U and U_R in the plane, they look exactly the same.

12 00:01:04,971 –> 00:01:09,476 In other words the open domain here does not make any problems at all.

13 00:01:10,300 –> 00:01:17,689 Ok, now in the last video we have learned that we can split up the function f into a real and an imaginary part.

14 00:01:18,343 –> 00:01:24,493 and both we can see as a function from U_R as a subset of R^2 into R.

15 00:01:24,886 –> 00:01:29,903 In particular the real part function here, we denote by a lower case u.

16 00:01:30,414 –> 00:01:34,057 So please recall this u is a real-valued function.

17 00:01:34,971 –> 00:01:41,053 and then exactly in the same way, the imaginary part of f gives us a function we call v.

18 00:01:41,414 –> 00:01:46,513 So also here we have a function with domain U_R and codomain R.

19 00:01:47,443 –> 00:01:53,299 Hence you see, the whole information of the function f is now translated into these 2 functions.

20 00:01:54,071 –> 00:02:02,086 Indeed the property that f is holomorphic translates into the fact that u and v fulfill the Cauchy-Riemann equations.

21 00:02:03,014 –> 00:02:07,826 These are not complicated at all. There are just 2 partial differential equations.

22 00:02:08,557 –> 00:02:17,597 The first one simply says that the partial derivative of u with respect to x is the same as the partial derivative of v with respect to y.

23 00:02:18,100 –> 00:02:23,286 and then in the second equation you see, when you switch the roles there is a minus sign included.

24 00:02:24,386 –> 00:02:32,388 In summary what we need for a holomorphic function is that these 2 equations are fulfilled at each point in the set U_R.

25 00:02:33,457 –> 00:02:37,523 In fact this equivalence here, we have discussed in the last video.

26 00:02:38,043 –> 00:02:41,611 Now in this video I want to show you concrete examples.

27 00:02:42,400 –> 00:02:46,033 and I would say, let’s immediately start with a very simple one.

28 00:02:46,986 –> 00:02:53,767 So let’s take the complex function f from C to C. Given as the identity.

29 00:02:54,671 –> 00:02:57,951 Which means the number z is simply mapped to z again.

30 00:02:59,014 –> 00:03:06,656 Now, because we want to deal with the partial derivatives with respect to x and y, we have to write z as (x + iy).

31 00:03:07,786 –> 00:03:11,431 Hence we do this change on the left-hand side and on the right-hand side.

32 00:03:12,314 –> 00:03:19,045 and then you see, here the real part is our function u and the imaginary part is our function v.

33 00:03:19,929 –> 00:03:23,830 So you see in this case it’s not complicated at all.

34 00:03:24,500 –> 00:03:28,929 and now we just want to check if the Cauchy-Riemann equations are satisfied.

35 00:03:29,957 –> 00:03:32,489 So let’s write down the partial derivatives.

36 00:03:33,300 –> 00:03:37,928 First we have du/dx. Which is in this case simply 1.

37 00:03:38,729 –> 00:03:47,597 In the same way we also see that dv/dy is very easy to calculate, because this is a simple function and the derivative is just 1 again.

38 00:03:48,771 –> 00:03:53,901 Hence we immediately see that the first equation is satisfied at all points.

39 00:03:54,271 –> 00:03:58,434 So let’s go to the second equation, where we have du/dy.

40 00:03:59,514 –> 00:04:03,578 There we see, there is no y in u. Therefore we get 0.

41 00:04:04,271 –> 00:04:08,837 In fact the same holds for dv/dx, because there is no x in v.

42 00:04:09,371 –> 00:04:12,459 and also of course, the minus sign does not change anything.

43 00:04:13,600 –> 00:04:17,681 In summary, both Cauchy-Riemann equations are fulfilled.

44 00:04:18,486 –> 00:04:22,290 From that we then can conclude that the function f is holomorphic.

45 00:04:23,429 –> 00:04:26,046 Of course this is a fact we have already known.

46 00:04:26,857 –> 00:04:33,745 Therefore I would say in the next example, let’s look at a function where we already know that it is not holomorphic.

47 00:04:34,571 –> 00:04:37,996 Indeed this was simply the complex conjugation.

48 00:04:38,814 –> 00:04:40,470 So f(z) is z bar.

49 00:04:41,500 –> 00:04:47,868 Now as before, in order to apply the Cauchy-Riemann equations we have to rewrite that with x and y.

50 00:04:48,586 –> 00:04:53,698 Of course this is not so complicated. The imaginary part just gets a minus sign.

51 00:04:54,743 –> 00:04:59,862 and then as before, the real part is u and the imaginary part here is v.

52 00:05:00,471 –> 00:05:05,664 So you see it’s different from before and it will change the Cauchy-Riemann equation.

53 00:05:06,771 –> 00:05:13,054 Of course here du/dx is still 1, but dv/dy is now -1.

54 00:05:13,914 –> 00:05:17,401 Therefore we can conclude: this is not the same.

55 00:05:18,057 –> 00:05:20,301 Indeed no matter which point we put in.

56 00:05:21,043 –> 00:05:26,498 In conclusion we get the result, we’ve expected. Namely f is not holomorphic.

57 00:05:27,500 –> 00:05:33,013 Here you see, showing this was very simple by just using the Cauchy-Riemann equations.

58 00:05:33,943 –> 00:05:38,022 However maybe we should also look at a more complicated example.

59 00:05:39,014 –> 00:05:42,099 So let’s take another complex function f.

60 00:05:42,814 –> 00:05:47,326 and now I want to look at it as z^2 + iz.

61 00:05:47,871 –> 00:05:54,400 To be honest. You already know that this is a holomorphic function, because it is a complex polynomial.

62 00:05:55,200 –> 00:06:01,485 Indeed we already know how to calculate the complex derivative for such a complex polynomial.

63 00:06:02,257 –> 00:06:05,101 Here it should be 2 times z + i.

64 00:06:05,543 –> 00:06:09,531 Still it’s helpful to check it with the Cauchy-Riemann equations.

65 00:06:10,586 –> 00:06:16,479 Now, by writing x + iy instead of z, we actually have something to do now,

66 00:06:17,057 –> 00:06:23,119 because in this expression here, we don’t see immediately the real and imaginary part.

67 00:06:24,100 –> 00:06:26,876 We first have to calculate a little bit.

68 00:06:27,500 –> 00:06:32,139 However that’s not so complicated, we simply can do the whole calculation here.

69 00:06:32,614 –> 00:06:42,871 So we have x^2 + i2xy - y^2 from the binomial + ix and -y.

70 00:06:43,729 –> 00:06:48,299 Now let’s put the real part to the left and the imaginary part to the right.

71 00:06:49,014 –> 00:06:55,322 So the real part is everything without an “i” and for the imaginary part I factor out one “i”.

72 00:06:56,457 –> 00:06:59,190 and then we have the result we wanted,

73 00:07:00,057 –> 00:07:05,038 because again, here we have our function u and there we have the function v.

74 00:07:05,986 –> 00:07:10,214 Now with these 2 functions we can check the Cauchy-Riemann equations again.

75 00:07:11,386 –> 00:07:15,814 First du/dx, not hard to see, is simply 2 times x

76 00:07:16,800 –> 00:07:23,738 Then the second partial derivative we want to calculate is dv/dy, which is also 2x.

77 00:07:24,857 –> 00:07:28,398 Hence our first equation is actually fulfilled.

78 00:07:29,000 –> 00:07:37,122 Going to the second one, you see we have to do a little bit more, because this derivative is -2y - 1

79 00:07:37,714 –> 00:07:44,592 Ok, then the last derivative we want to calculate would be the partial derivative of v with respect to x.

80 00:07:45,457 –> 00:07:48,741 and there you see, it’s 2y + 1

81 00:07:49,543 –> 00:07:53,746 and then for our equation we just have to put a minus in front.

82 00:07:54,229 –> 00:07:58,317 and then we recognize this is the same as du/dy.

83 00:07:59,071 –> 00:08:04,858 Therefore also here both equations are fulfilled. So the function f is holomorphic.

84 00:08:05,514 –> 00:08:08,421 I already told you, this is not a surprise for us.

85 00:08:09,429 –> 00:08:14,871 However with this example you have learned how you can apply the Cauchy-Riemann equations,

86 00:08:14,914 –> 00:08:18,857 when the function f is given as a real and an imaginary part.

87 00:08:19,820 –> 00:08:25,952 and in fact you also get the complex derivative of f from these partial derivatives here,

88 00:08:26,757 –> 00:08:30,069 but I think this is a good topic for the next video.

89 00:08:30,757 –> 00:08:36,457 Therefore I really hope that I see you there. Have a nice day and bye!

Q1: A complex function $f: \mathbb{C} \rightarrow \mathbb{C}$ is holomorphic if and only if

A1: the real part $u$ of $f$ and the imaginary part of $v$ of $f$ satisfy the Cauchy-Riemann equations at all points.

A2: the real part $u$ of $f$ and the imaginary part of $v$ of $f$ satisfy the Cauchy-Riemann equations at one point.

A3: the real part $u$ of $f$ and the imaginary part of $v$ of $f$ don’t satisfy the Cauchy-Riemann equations at all points.

A4: the real part $u$ of $f$ and the imaginary part of $v$ of $f$ don’t satisfy the Cauchy-Riemann equations at one point.

Q2: Consider the complex function $f: \mathbb{C} \rightarrow \mathbb{C}$ given by $f(z) = |z|^2 $. Are the Cauchy-Riemann equations satisfied at $z_0 = 0$?

A1: Yes!

A2: No!

Q3: Consider again the complex function $f: \mathbb{C} \rightarrow \mathbb{C}$ given by $f(z) = |z|^2 $. Is the function holomorphic?

A1: Yes, the Cauchy-Riemann equations are satisfied.

A2: No, the Cauchy-Riemann equations are not satisfied everywhere.