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Title: Complex Derivative and Examples
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Series: Complex Analysis
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YouTube-Title: Complex Analysis 3 | Complex Derivative and Examples
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Bright video: https://youtu.be/_Q4pGe6GyR4
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Dark video: https://youtu.be/00-aqxRoOug
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Ad-free video: Watch Vimeo video
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Quiz: Test your knowledge
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Subtitle on GitHub: ca03_sub_eng.srt
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Timestamps
00:00 Intro
00:34 The [geometric] intuition for complex derivative
04:11 Producing the formal definition
05:19 Example 1: A linear polynomial in ℂ
07:34 Example 2: A conjugate function
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Subtitle in English
1 00:00:00,600 –> 00:00:04,035 Hello and welcome back to complex analysis.
2 00:00:04,886 –> 00:00:11,628 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 00:00:12,543 –> 00:00:17,617 Now, today in part 3 we continue talking about complex differentiability.
4 00:00:18,071 –> 00:00:25,610 More precisely we introduce the complex derivative and I’ll explain to you why this gives us a linear approximation.
5 00:00:26,186 –> 00:00:30,358 and finally at the end of the video, I show you concrete examples.
6 00:00:31,100 –> 00:00:38,708 However first please recall that the domain of the function we consider is always an open set, we call U.
7 00:00:39,114 –> 00:00:44,086 Moreover from this set U we always fix a point z_0.
8 00:00:45,186 –> 00:00:50,018 Now, because this set is open, z_0 is always an inner point.
9 00:00:50,471 –> 00:00:55,518 Which means that there are a lot of points around it, such that limits makes sense.
10 00:00:56,571 –> 00:01:03,200 Indeed this is what we need, when we want to define that the function f is differentiable at the point z_0.
11 00:01:04,057 –> 00:01:10,810 and if we want to make it clear that we have a complex function here, we say complex differentiable at z_0.
12 00:01:11,829 –> 00:01:18,339 Now, we have seen in the last video that this definition here looks exactly the same as for real functions.
13 00:01:18,539 –> 00:01:21,069 Namely, that the following limit exists.
14 00:01:21,943 –> 00:01:26,298 Indeed this is the slope of the function at a given point.
15 00:01:26,498 –> 00:01:29,670 and often it is called the differentiable quotient.
16 00:01:30,457 –> 00:01:36,021 Now since this describes the slope, we can reformulate it to a linear approximation.
17 00:01:36,529 –> 00:01:42,243 The only problem is that the visualization is not so good when we deal with complex numbers.
18 00:01:43,214 –> 00:01:48,130 Here I mean that for a real valued function, you can simply draw the graph of the function.
19 00:01:48,771 –> 00:01:50,856 So maybe it could look like this.
20 00:01:51,714 –> 00:01:58,258 However for a complex function the input is the complex plane and the output is the complex plane.
21 00:01:58,757 –> 00:02:04,119 Hence the correct visualization of the graph of the function should live in 4 dimensions.
22 00:02:04,943 –> 00:02:08,598 So you see there is no way I can correctly draw this here.
23 00:02:09,414 –> 00:02:15,719 Hence we just take an abstract picture then, where the x-axis represents the whole complex plane C.
24 00:02:16,557 –> 00:02:22,157 and in the same way the output, the y-axis also represents the whole complex plane.
25 00:02:22,637 –> 00:02:25,429 So you see we lose a lot of information here.
26 00:02:25,443 –> 00:02:29,457 It’s not the best picture at all, but we can see the linear approximation then.
27 00:02:30,529 –> 00:02:36,540 In other words we approximate the function f around this point z_0 with a linear function.
28 00:02:37,314 –> 00:02:41,847 Therefore you recognize, this works exactly as for real-valued functions.
29 00:02:42,700 –> 00:02:48,979 So we get a statement that is equivalent to our definition of being differentiable at the point z_0.
30 00:02:49,557 –> 00:02:54,644 Namely we just have to give this expression a new name and we call it a function delta.
31 00:02:55,543 –> 00:03:00,982 and this delta gets an index, where we write the function f and the point z_0.
32 00:03:01,643 –> 00:03:07,598 Then you see z could be any point from U, so we have a function from U to C.
33 00:03:08,171 –> 00:03:16,182 Ok and now we can just reformulate this expression here and we get the property the function delta_f,z_0 should have.
34 00:03:16,543 –> 00:03:23,858 Namely f(z) can be written as the constant f(z_0), so the value at the point z_0
35 00:03:24,714 –> 00:03:28,098
- a linear term times the delta function.
36 00:03:28,757 –> 00:03:35,181 Of course this property here holds for every function f. So we need one more ingredient here.
37 00:03:35,971 –> 00:03:39,610 and here please recall, we want that this limit exists.
38 00:03:40,557 –> 00:03:47,084 Hence this means that the function delta_f,z_0 is continuous at the point z_0.
39 00:03:47,743 –> 00:03:53,945 Hence this continuity property is really what we need and this one makes the differentiability.
40 00:03:55,000 –> 00:04:02,114 Now, with this linear part here you might already see why we call this a linear approximation of the function f.
41 00:04:02,886 –> 00:04:09,509 If you have never seen this, you can watch my real analysis course, where I talk a lot about linear approximations.
42 00:04:10,371 –> 00:04:14,666 However here, we finally should define the derivative of the function f.
43 00:04:15,400 –> 00:04:19,177 This not so complicated, because now we have everything.
44 00:04:19,729 –> 00:04:24,698 Moreover you might already know the derivative is denoted with f'.
45 00:04:25,114 –> 00:04:28,671 and then in parentheses we write the point z_0.
46 00:04:29,386 –> 00:04:38,002 and now by definition the value of the derivative is simply the output of the delta function we get, when we put z_0 into it.
47 00:04:38,614 –> 00:04:46,865 and now because delta_f,z_0 is a continuous function at the point z_0, we can calculate this number with a limit.
48 00:04:47,529 –> 00:04:49,623 Namely, simply this limit here.
49 00:04:50,629 –> 00:04:53,712 Or in other words, it’s simply the same thing.
50 00:04:54,314 –> 00:05:00,074 and now this complex number we call the derivative of f at the point z_0.
51 00:05:00,829 –> 00:05:09,533 and again if you want to make it clear that we work with complex functions, you speak of the complex derivative of f at z_0.
52 00:05:10,100 –> 00:05:18,363 So you see the definition is not complicated at all, but please always keep in mind: f’(z_0) is a complex number.
53 00:05:19,200 –> 00:05:23,186 Ok, then I think we are ready to look at 2 simple examples.
54 00:05:24,357 –> 00:05:30,183 The first one we consider should always be the linear polynomial, but now the complex valued one.
55 00:05:30,943 –> 00:05:36,643 So we have a function f, where the domain U can be chosen as the whole complex plane C.
56 00:05:36,971 –> 00:05:41,475 and the definition of this complex function would be f(z)
57 00:05:42,343 –> 00:05:48,419 is equal to a complex number m times z + another complex number c.
58 00:05:49,214 –> 00:05:54,092 So this is simply the common linear polynomial, but now complex valued.
59 00:05:54,986 –> 00:06:00,800 There, because the function is already linear, a linear approximation will not change so much.
60 00:06:01,357 –> 00:06:07,418 However let’s try for a given z_0 to rewrite the function in this form.
61 00:06:07,957 –> 00:06:11,043 Hence first we need f(z_0).
62 00:06:11,614 –> 00:06:16,087 Which is not so complicated, because it’s simply m times z_0 + c.
63 00:06:16,943 –> 00:06:22,037 and then we want to add (z - z_0) times something.
64 00:06:22,529 –> 00:06:26,559 Now of course this something we now can figure out.
65 00:06:27,400 –> 00:06:31,029 What we want to get in the end here is m times z + c
66 00:06:31,471 –> 00:06:35,757 The c we have already got, so we have to get rid of this term here.
67 00:06:36,986 –> 00:06:41,393 Of course this is no problem for us, because we can multiply this term by m.
68 00:06:42,300 –> 00:06:48,284 Then z_0 times m will cancel out and the only thing that remains is z times m.
69 00:06:48,484 –> 00:06:52,214 Or in other words, now we have rewritten the function.
70 00:06:52,800 –> 00:06:58,659 The first term is f(z_0) and this part here is our function delta.
71 00:06:59,214 –> 00:07:05,329 The function is constant. It’s equal to m, so we don’t have any problem putting z_0 into it.
72 00:07:06,371 –> 00:07:10,855 Hence we know the derivative, no matter which point z_0 we choose.
73 00:07:11,329 –> 00:07:14,489 The derivative is always equal to m.
74 00:07:15,157 –> 00:07:20,370 This shouldn’t be a surprise to you, because the m stands for the slope of the linear function here.
75 00:07:21,143 –> 00:07:26,819 However still now we have proven that this function is indeed complex differentiable.
76 00:07:27,019 –> 00:07:30,814 and indeed no matter which point z_0 we consider.
77 00:07:31,786 –> 00:07:36,752 Ok, for the next part let’s look at an example that a function is not differentiable.
78 00:07:37,371 –> 00:07:41,057 Again let’s take the domain as the whole complex plane.
79 00:07:41,529 –> 00:07:47,355 and now the definition of the function should be that z is sent to the complex conjugate of z.
80 00:07:47,929 –> 00:07:51,480 So with our notation we would right z bar.
81 00:07:52,514 –> 00:07:56,041 So you see this is not a complicated function at all.
82 00:07:56,557 –> 00:08:01,966 To see what the function does, let’s draw the input space here and the output space here.
83 00:08:02,657 –> 00:08:06,425 So our function f maps from the left-hand side to the right-hand side.
84 00:08:07,286 –> 00:08:16,270 Now let’s take any point z from the domain and then the function f just flips this point with respect to the x-axis.
85 00:08:17,000 –> 00:08:22,164 The real part of the numbers stays the same, but the imaginary part gets a different sign.
86 00:08:22,986 –> 00:08:27,257 So you see this reflection is all the function f does.
87 00:08:27,914 –> 00:08:33,041 Therefore it’s not hard to show that this function is indeed a continuous function.
88 00:08:33,857 –> 00:08:40,364 However it turns out that it’s not differentiable. No matter at which point z_0 we look.
89 00:08:41,029 –> 00:08:48,452 and this is immediately interesting, because in real analysis it’s not so simple to find such a strange function.
90 00:08:49,543 –> 00:08:53,671 Indeed in complex analysis this property is not strange at all.
91 00:08:54,557 –> 00:09:02,080 In order to see this I would suggest that we try calculating the derivative at the point z_0 = 0.
92 00:09:02,771 –> 00:09:06,646 Indeed all the other points will work exactly the same.
93 00:09:07,443 –> 00:09:14,265 Now by definition, if the derivative exists f’(0) is given by our limit.
94 00:09:14,743 –> 00:09:20,289 So we have f(z) - f(0) divided by (z - 0).
95 00:09:20,914 –> 00:09:26,537 So we can immediately simplify this, because when we when we flip 0, we don’t change anything.
96 00:09:27,243 –> 00:09:32,276 Hence we have the limit z to 0 of z bar divided by z.
97 00:09:32,943 –> 00:09:38,141 Ok and now it turns out that exactly this limit does not exist.
98 00:09:39,157 –> 00:09:47,343 In order to see this please recall that z to 0 here means that in a complex plane we have a lot of possibilities.
99 00:09:48,257 –> 00:09:51,600 So we can go from different directions into 0.
100 00:09:52,329 –> 00:09:58,049 For example we could use the real number line to go from the right hand side into 0.
101 00:09:58,543 –> 00:10:03,758 For example with the sequence z_n = 1 over n.
102 00:10:04,643 –> 00:10:10,071 Hence there the question is: what is z_n bar divided by z_n?
103 00:10:10,486 –> 00:10:16,302 Of course this is not so hard to see. This is 1 over n divided by 1 over n.
104 00:10:17,086 –> 00:10:21,532 Or in other words, it does not depend on n. It’s always the same number, 1.
105 00:10:22,214 –> 00:10:26,754 Therefore also in the limit n to infinity, we get out 1.
106 00:10:27,429 –> 00:10:32,111 So the conclusion is, if this limit exists it has to be 1.
107 00:10:32,686 –> 00:10:36,750 That this can’t be true we see, when we look in another direction.
108 00:10:37,943 –> 00:10:42,882 Now let’s use the imaginary axis and one from the bottom to 0.
109 00:10:43,386 –> 00:10:50,652 For example this can be done with the sequence z_n is equal to -i divided by n.
110 00:10:51,429 –> 00:10:57,343 So again we have the same question. What is z_n bar divided by z_n?
111 00:10:57,877 –> 00:11:02,026 So now in the numerator we actually have a flip.
112 00:11:03,029 –> 00:11:09,246 It’s simply +i divided by n divided -i divided by n.
113 00:11:10,814 –> 00:11:15,508 Therefore we can cancel everything and the only thing that remains is the - sign.
114 00:11:15,708 –> 00:11:17,155 So we get out -1.
115 00:11:17,900 –> 00:11:22,282 So we see, also in the limit n to infinity this is -1.
116 00:11:23,557 –> 00:11:28,225 Which is of course not the same as before. So this is not equal.
117 00:11:28,971 –> 00:11:33,841 Therefore the conclusion is: The limit from before does not exist.
118 00:11:34,857 –> 00:11:40,778 In other words f is not differentiable at the point z_0 = 0.
119 00:11:41,271 –> 00:11:46,008 Accordingly we immediately have a counterexample for differentiability.
120 00:11:46,957 –> 00:11:52,956 Here please recall I already told you, you can do the same prove for every other point z_0.
121 00:11:53,629 –> 00:11:58,646 So the conclusion is indeed, the function is nowhere complex differentiable.
122 00:11:59,043 –> 00:12:05,779 At first this might be surprising, because the function is continuous and indeed a very simple function.
123 00:12:06,743 –> 00:12:14,586 In fact we can reformulate this and say that differentiability for complex functions is a very strong property.
124 00:12:15,429 –> 00:12:18,787 and I can promise you, this will get very apparent soon.
125 00:12:19,543 –> 00:12:22,830 and with this I hope I see you in the next video.
126 00:12:23,030 –> 00:12:23,957 Bye!
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Quiz Content
Q1: What is not a correct definition for a function $f: \mathbb{C} \rightarrow \mathbb{C}$ being complex differentiable at $z_0 \in \mathbb{C}$?
A1: $\displaystyle \lim_{z \rightarrow z_0} \frac{f(z) - f(z_0)}{z - z_0}$ exists.
A2: There is a function $\Delta_{f,z_0} : \mathbb{C} \rightarrow \mathbb{C}$ that is continuous at $z_0$ and satifies $$f(z) = f(z_0) + (z - z_0) \cdot \Delta_{f,z_0}(z)$$ for all $z \in \mathbb{C}$.
A3: There is a number $b \in \mathbb{C}$ and a function $r : \mathbb{C} \rightarrow \mathbb{C}$ that is continuous at $z_0$ with $r(z_0) = 0$ and satifies $$f(z) = f(z_0) + (z - z_0) \cdot b + (z - z_0) \cdot r(z)$$ for all $z \in \mathbb{C}$.
A4: $f$ is continuous at $z_0$ and the limit $\displaystyle \lim_{z \rightarrow z_0} \frac{f(z-z_0) - f(z_0)}{z - z_0}$ exists.
Q2: What is derivative of the function $f: \mathbb{C} \rightarrow \mathbb{C}$ given by $f(z) = 3 z + 2$?
A1: $f(z_0) = 2$
A2: $f(z_0) = 3$
A3: $f(z_0) = 3 z_0$
A4: $f(z_0) = 2 z_0$
A5: $f(z_0) = 0$
Q3: Which of the following limits does not exist?
A1: $\displaystyle \lim_{z \rightarrow 0} z$
A2: $\displaystyle \lim_{z \rightarrow 0} \overline{z} $
A3: $\displaystyle \lim_{z \rightarrow 0} \frac{|z|^2}{z} $
A4: $\displaystyle \lim_{z \rightarrow 0} \frac{\overline{z}}{z} $
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Last update: 2024-10
