00:00 Intro

00:19 Is differentiation in ℂ and ℝ² the same?

01:43 Each complex function induces a real function

06:01 When is a map totally differentiable?

10:11 The meaning of Jacobian matrix and example

1 00:00:00,600 –> 00:00:03,823 Hello and welcome back to complex analysis.

2 00:00:04,914 –> 00:00:12,403 and of course as always, first I want to thank all the nice people that support this channel on Steady, via Paypal or by other means.

3 00:00:13,157 –> 00:00:19,630 Now in today’s part 5 we go into the direction of the Cauchy-Riemann equations of complex analysis.

4 00:00:20,586 –> 00:00:26,690 In order to understand them, we first have to talk about differentiability in higher dimensions.

5 00:00:27,343 –> 00:00:34,404 In particular we need to know the notion of total differentiability from functions from R^2 into R^2.

6 00:00:34,929 –> 00:00:41,646 The reason for that you immediately recognize, when we visualize the complex numbers as the complex plane.

7 00:00:42,329 –> 00:00:47,254 So you know, a complex number has a real part and an imaginary part.

8 00:00:48,229 –> 00:00:56,035 Therefore the natural question you could ask here is: is there any difference between this and the vector space R^2?

9 00:00:56,700 –> 00:00:59,700 At first glance both totally look the same.

10 00:01:00,429 –> 00:01:06,597 and even the operation how we add vectors and add complex numbers is actually the same.

11 00:01:07,200 –> 00:01:12,445 In fact if you know the construction of the complex numbers, this is no surprise for you.

12 00:01:13,243 –> 00:01:18,829 For this we know that C is R^2, but it carries one more operation.

13 00:01:19,857 –> 00:01:24,933 and this is the multiplication we have to add to the vector space to get the complex numbers.

14 00:01:25,629 –> 00:01:34,929 This all is important to know, because it means, when we define a map from C to C, it’s the same as defining a map from R^2 to R^2.

15 00:01:35,957 –> 00:01:41,112 Therefore it’s important to understand how we can switch between both cases.

16 00:01:41,543 –> 00:01:46,714 So let’s remember the fact that each complex function induces a real function.

17 00:01:47,629 –> 00:01:54,699 More precisely, if we have a map f from C to C, we can also define a map from R^2 to R^2.

18 00:01:55,286 –> 00:02:00,703 and maybe for the moment let’s call this map on the right-hand side f with an index R.

19 00:02:01,714 –> 00:02:04,757 Ok, of course this also works the other way around.

20 00:02:04,837 –> 00:02:10,935 So if you start with a map f_R here, we can define a map f from C to C.

21 00:02:11,586 –> 00:02:18,637 Now, how we exactly do this, we could formally write down, but maybe it’s better to just look at an example.

22 00:02:19,471 –> 00:02:23,171 I really think after seeing this example, everything is clear.

23 00:02:24,157 –> 00:02:28,126 Therefore the function we consider now shouldn’t be too complicated.

24 00:02:29,500 –> 00:02:34,652 Indeed just let’s look at a quadratic function, which sends z to z^2.

25 00:02:35,529 –> 00:02:39,167 Now, squaring a complex number is not so hard.

26 00:02:39,757 –> 00:02:44,431 In order to get a visualization for the function, let’s draw the complex plane again.

27 00:02:45,443 –> 00:02:50,727 Now, the function f maps the complex plane from the left-hand side to the right-hand side.

28 00:02:51,857 –> 00:02:55,771 For example if you consider the point “i” here one the left.

29 00:02:56,629 –> 00:03:01,129 Then it simply gets squared and we find it as -1 on the right.

30 00:03:02,286 –> 00:03:07,238 In the same way, you could take any other point to visualize how it gets mapped.

31 00:03:08,314 –> 00:03:15,229 Now, on the calculation side of the problem, you could take the real and the imaginary part of z and calculate with them.

32 00:03:16,129 –> 00:03:24,064 Hence we could say x + iy is mapped to (x + iy)^2.

33 00:03:25,043 –> 00:03:28,897 and then, without a problem we can just expand this expression.

34 00:03:29,857 –> 00:03:37,998 So we get x^2 + 2ixy - y^2.

35 00:03:38,814 –> 00:03:47,435 Of course this now looks more complicated than just z^2, but it now tells us how we can define the map f_R.

36 00:03:47,971 –> 00:03:54,510 We simply know and use that the real part is the first coordinate and the imaginary part is the second coordinate.

37 00:03:55,443 –> 00:04:02,275 More precisely we write f_R from R^2 into R^2 and we start with a vector with 2 coordinates.

38 00:04:03,000 –> 00:04:07,080 As we said, the first one is x and the second one is y.

39 00:04:07,700 –> 00:04:11,896 and this vector now gets mapped to a new vector with 2 components.

40 00:04:13,014 –> 00:04:19,092 Now, the first component should be the real part of this. Which means x^2 - y^2.

41 00:04:19,971 –> 00:04:26,001 In the same way, the second component should be the imaginary part. Which means everything with an “i”.

42 00:04:27,000 –> 00:04:32,471 Ok, now we have the map f_R. Which carries exactly the same information as f.

43 00:04:33,500 –> 00:04:38,029 Indeed we can simply check that with the same example we have given with the picture.

44 00:04:39,086 –> 00:04:44,597 Hence, here we would put in the vector (0, 1). Which represents this point.

45 00:04:45,529 –> 00:04:55,082 Now, using our definition of f_R, we can calculate and get in the first component -1 and in the second component 0.

46 00:04:55,971 –> 00:05:00,896 and by seeing this you know, this vector is to be found here.

47 00:05:01,914 –> 00:05:08,300 Ok, in summary this shows us, that if we want to analyse a complex function f,

48 00:05:08,500 –> 00:05:13,759 we can equivalently analyze a real function from R^2 into R^2.

49 00:05:14,571 –> 00:05:19,996 However in this connection it’s not clear at all what differentiability means.

50 00:05:20,614 –> 00:05:27,056 We have a definition for complex functions, which means we also need one for such functions here.

51 00:05:28,286 –> 00:05:36,467 Indeed you might already know that differentiability makes totally sense for functions between R^n and R^m.

52 00:05:37,171 –> 00:05:45,327 So you see the exact dimensions for the vector spaces here and there are not important for the definition of differentiability.

53 00:05:45,800 –> 00:05:50,843 and exactly for this reason the notion for differentiability for these functions

54 00:05:51,114 –> 00:05:55,200 is different from the one we already know for complex functions.

55 00:05:56,114 –> 00:06:01,185 Therefore our task here is to form the connection between both notions.

56 00:06:01,986 –> 00:06:10,201 However before we can do that I should first explain to you how we define the derivative for functions between vector spaces.

57 00:06:11,200 –> 00:06:16,331 Of course for us it’s sufficient to only consider maps between R^2 and R^2.

58 00:06:16,743 –> 00:06:22,778 Nevertheless you will see this definition works, no matter which R^n and R^m we choose.

59 00:06:23,543 –> 00:06:29,455 Now, the correct notion we now introduce for such maps is also called differentiable.

60 00:06:30,329 –> 00:06:35,306 However sometimes it’s needed to distinguish it from all the other differentiability notions

61 00:06:35,506 –> 00:06:39,142 and therefore it’s often called totally differentiable.

62 00:06:40,186 –> 00:06:43,200 Ok, so the idea here is the same as always.

63 00:06:43,400 –> 00:06:46,917 We want a linear approximation around a given point.

64 00:06:47,886 –> 00:06:55,869 Therefore we have to fix a point in R^2. So maybe let’s simply call it (x_0, y_0).

65 00:06:56,786 –> 00:07:04,057 and now you should know a linear map in R^2 for the linear approximation can be described by a 2x2 matrix.

66 00:07:05,200 –> 00:07:08,500 and this matrix we want to call J.

67 00:07:09,100 –> 00:07:15,646 Of course everything is real here. So the entries of the matrix J are also real numbers.

68 00:07:16,429 –> 00:07:21,381 Ok, with this we are ready to write down what we mean by linear approximation here.

69 00:07:22,543 –> 00:07:30,630 The function f_R at any point (x, y) can be written as a constant + a linear term + an error term.

70 00:07:31,429 –> 00:07:37,232 and indeed this error term should be small, when (x, y) is close to (x_0, y_0).

71 00:07:38,214 –> 00:07:45,100 So more concretely first the constant is the value of f_R at the position (x_0, y_0)

72 00:07:46,171 –> 00:07:49,799 and then comes the linear map. Which is given by the matrix J.

73 00:07:50,714 –> 00:07:57,621 To get a linear map, this matrix is multiplied from the right-hand side by the difference of the 2 vectors.

74 00:07:58,486 –> 00:08:01,825 So we have (x, y) - (x_0, y_0).

75 00:08:02,514 –> 00:08:07,240 and with this we already have what we call the linear approximation.

76 00:08:08,214 –> 00:08:14,942 Therefore the only thing we have append now is the error term. Which measures how good this approximation is.

77 00:08:15,986 –> 00:08:19,358 and this one is given by a map we call phi.

78 00:08:20,143 –> 00:08:25,729 Of course here the input and the output space should be the same as for f_R.

79 00:08:26,657 –> 00:08:34,977 Now, the important claim here is that this error term gets smaller and smaller, when (x, y) gets to our point (x_0, y_0).

80 00:08:35,671 –> 00:08:40,836 Indeed then the term should go to 0 and faster than linearly.

81 00:08:41,671 –> 00:08:47,786 Namely this means when we look at phi(x, y) and divide it by the length of the difference vector

82 00:08:47,929 –> 00:08:51,552 So the vector (x, y) - (x_0, y_0)

83 00:08:52,186 –> 00:08:57,455 Then this expression should go to 0 when (x, y) goes to (x_0, y_0).

84 00:08:58,414 –> 00:09:02,836 Ok so this is what we need, but maybe I should explain some details here.

85 00:09:03,686 –> 00:09:08,852 First this double line here we use for the notation of a norm in a vector space.

86 00:09:09,743 –> 00:09:13,743 Therefore with such a norm we just measure the length of a vector.

87 00:09:14,529 –> 00:09:19,350 and most of the time in R^2 this is just the standard euclidean norm.

88 00:09:20,214 –> 00:09:25,101 and this one you should know. It’s simply the square root of x^2 + y^2.

89 00:09:26,043 –> 00:09:31,980 Now, in addition when we can measure lengths in the vector space, we can also measure distances.

90 00:09:32,886 –> 00:09:37,774 Therefore it makes sense that we say that one vector converges to another vector.

91 00:09:38,871 –> 00:09:43,246 It simply means that the length of the difference vector goes to 0.

92 00:09:44,157 –> 00:09:49,729 Therefore you see this whole definition also makes sense when we are in another vector space

93 00:09:49,771 –> 00:09:54,071 like R^3, R^4 or just the vector space where we can measure lengths.

94 00:09:55,057 –> 00:09:58,071 and exactly for this reason and please keep that in mind,

95 00:09:58,357 –> 00:10:03,914 this definition is not the same as the differentiability definition for the complex numbers.

96 00:10:05,043 –> 00:10:13,670 However maybe you are very familiar with this definition here and in this case I can tell you, this matrix J has special name.

97 00:10:14,257 –> 00:10:18,004 Most of the time it’s simply called the Jacobian matrix.

98 00:10:18,843 –> 00:10:25,323 and more precisely we would say, Jacobian matrix of f_R at the position (x_0, y_0).

99 00:10:26,443 –> 00:10:33,399 Now, if you are used to partial derivatives, you might also know that the Jacobian matrix has a very nice representation.

100 00:10:34,414 –> 00:10:39,525 Please recall, in our case it’s a 2x2 matrix. So it has 4 entries.

101 00:10:40,514 –> 00:10:48,206 Now, in the first column you find the partial derivative of f_R with respect to the first variable. Which is x.

102 00:10:49,171 –> 00:10:52,442 Here please don’t forget, this is a vector with 2 components.

103 00:10:53,586 –> 00:10:56,985 So in fact, we have 2 entries of the matrix here.

104 00:10:57,871 –> 00:11:06,568 Ok and maybe not a surprise for you, the second column is given by the partial derivative of f_R with respect to the second coordinate.

105 00:11:07,857 –> 00:11:11,887 So in other words we have the partial derivative with respect to y.

106 00:11:12,943 –> 00:11:17,074 and again this is a whole vector. So we get 2 entries of the matrix.

107 00:11:18,186 –> 00:11:24,524 Now of course in our case here, you would evaluate these partial derivatives at the point (x_0, y_0).

108 00:11:25,629 –> 00:11:30,182 and then we just have a 2x2 matrix, where the entries are real numbers.

109 00:11:31,529 –> 00:11:38,869 Ok, now in the case you have never seen this and this is all confusing for you, I really think an example can help.

110 00:11:39,429 –> 00:11:43,121 With this example you should see, it’s not so complicated after all.

111 00:11:44,271 –> 00:11:48,283 and for consistency let’s reconsider the example from above.

112 00:11:49,286 –> 00:11:56,048 So this is a nice function. We have x and y as the input and the output has 2 components as well.

113 00:11:56,814 –> 00:12:03,950 Then in the next step we can just calculate this Jacobian matrix J and show that we indeed have a linear approximation.

114 00:12:04,814 –> 00:12:09,686 Indeed the result will be this function here is differentiable at all points.

115 00:12:11,014 –> 00:12:14,151 Therefore let’s simply look at the partial derivatives here.

116 00:12:15,343 –> 00:12:22,388 So first we just form the partial derivative with respect to x. Which means here y is just a constant.

117 00:12:23,414 –> 00:12:30,387 Therefore in the first component we get 2 times x + 0, because y is a constant.

118 00:12:31,286 –> 00:12:35,012 and then in the second component we get 2 times y,

119 00:12:35,229 –> 00:12:39,398 because y is still a constant and the derivative of x is simply 1.

120 00:12:40,443 –> 00:12:43,414 Ok then for the second column let’s switch the roles.

121 00:12:43,486 –> 00:12:47,102 Now we form the partial derivative with respect to y.

122 00:12:47,957 –> 00:12:50,608 Or in other words, now x is a constant.

123 00:12:51,671 –> 00:12:56,967 For this reason the first component tells us, now we have -2 times y.

124 00:12:58,057 –> 00:13:03,699 So, this was just this part here and now the last step is just considering this function here.

125 00:13:04,814 –> 00:13:09,619 and the derivative with respect to y just gives us 2 times x.

126 00:13:10,343 –> 00:13:14,681 Ok so this is the Jacobian matrix of the function f_R.

127 00:13:15,714 –> 00:13:23,362 and here please recall, this function f_R represents a complex function f, which was complex differentiable.

128 00:13:24,029 –> 00:13:28,376 and now maybe you recognize some symmetries in this matrix here.

129 00:13:29,429 –> 00:13:33,363 Indeed you see this is a very special matrix we get out.

130 00:13:34,557 –> 00:13:43,090 With this I would say let’s use the next video to explain why this structure of the matrix is so important for complex analysis.

131 00:13:43,986 –> 00:13:46,968 Therefore I hope I see you there and have a nice day.

132 00:13:47,168 –> 00:13:47,943 Bye!

Q1: Consider the function $f_R: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ given by: $$ f \left( \binom{x}{y} \right) = \binom{x}{-y}$$ What is the corresponding complex function $f: \mathbb{C} \rightarrow \mathbb{C}$?

A1: $f(z) = z$

A2: $f(z) = \overline{z}$

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

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

A5: $f(z) = |z|$

Q2: Consider again the function $f_R: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ given by: $$ f \left( \binom{x}{y} \right) = \binom{x}{-y}$$ What is the Jacobian matrix of $f_R$ at position $\binom{x_0}{y_0}$?

A1: $$J = \begin{pmatrix} x_0 & x_0 \ x_0 & x_0 \end{pmatrix} $$

A2: $$J = \begin{pmatrix} 1 & x_0 \ 1 & y_0 \end{pmatrix} $$

A3: $$J = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} $$

A4: $$J = \begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix} $$

A5: $$J = \begin{pmatrix} 1 & 1 \ 1 & 1 \end{pmatrix} $$

A7: $$J = \begin{pmatrix} 1 & 1 \ -1 & 1 \end{pmatrix} $$

Q3: Consider the function $f: \mathbb{C} \rightarrow \mathbb{C}$ given by $f(z) = \overline{z} z $. Find the corresponding function $f_R: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ and determine the Jacobian matrix of $f_R$ at position $\binom{x_0}{y_0}$.

A1: $$J = \begin{pmatrix} 2 x_0 & 2 x_0 \ 2 x_0 & 2 x_0 \end{pmatrix} $$

A2: $$J = \begin{pmatrix} 2 x_0 & 2 y_0 \ 2 x_0 & 2 y_0 \end{pmatrix} $$

A3: $$J = \begin{pmatrix} 0 & 2 y_0 \ 0 & 2 y_0 \end{pmatrix} $$

A4: $$J = \begin{pmatrix} 2 x_0 & 2 y_0 \ 0 & 0 \end{pmatrix} $$

A5: $$J = \begin{pmatrix} 0 & 0 \ 2 x_0 & 2 y_0 \end{pmatrix} $$

A6: $$J = \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix} $$