The series is a series of differential equations 3Blue1Brown video notes, the original video can be seen: https://www.bilibili.com/video/av50290975 or https://www.youtube.com/watch?v=p_di4Zn4wz4&list=PLZHQObOWTQDNPOjrT6KVlfJuKtYTftqH6
Due to the limited level of the author, the text will inevitably some shortcomings and wrong, honesty please criticism.
1 Introduction
In 3B1B series of differential equations notes (three) we introduced to solve the heat conduction equation step two conditions, partial differential equations and boundary conditions themselves. We understand that the right solution can be used as a cosine function, but the temperature curve in reality often falls far short of the cosine function. Therefore, we need to fit a plurality of cosine function by means of a linear combination of the temperature profile, as a linear combination of the plurality of solutions is a new solution of the equation. This article will introduce a powerful way to fit - Fourier series. The article notes but does not contain all the knowledge points in the video, contain only a temporary part of the knowledge point I understand, so here is strongly recommended that you learn from the original video, or MA students of how to understand Fourier series formula .
2 Understanding and Solving
Fourier series like this long formula:
\ [\ the aligned the begin {} F (T) = & \ A_ FRAC {{0}}} + {2}. 1 A_ {\ COS (\ Omega T) + {B_ 1} \ sin (\ omega t ) \\ & + a_ {2} \ cos (2 \ omega t) + b_ {2} \ sin (2 \ omega t) \\ & + \ ldots \\ & = \ frac {a_ {0}} {2 } + \ sum_ {n = 1} ^ {\ infty} \ left [a_ {n} \ cos (n \ omega t) + b_ {n} \ sin (n \ omega t) \ right] \ end {aligned}
\] wherein:
\ [\ the begin {the aligned} A_ {n-} & = \ FRAC {2} {T} \ the int_ {T_ {0}} ^ {T_ {0} + T} f (t) \ cos (n \ omega t) dt \\ b_ {n} & = \ frac {2} {T} \ int_ {t_ {0}} ^ {t_ {0} + T} f (t) \ sin (n \ omega t)
dt \ end {aligned} \] essence: any periodic function or periodic signal into a set of (possibly infinite number of elements) of simple sine or cosine function.
Let's take an example:
\ [\ FRAC {. 4} {\ PI} \ left (\ FRAC {\ COS (. 1 \ PI X)} {. 1} - \ FRAC {\ COS (. 3 \ PI X)} {. 3 } + \ frac {\ cos ( 5 \ pi x)} {5} - \ frac {\ cos (7 \ pi x)} {7} + \ cdots \ right) = \ left \ {\ begin {array} { cl} 1 & \ text {if
} x <0.5 \\ 0 & \ text {if} x = 0.5 \\ - 1 & \ text {if} x>. 0.5 \ end {array} \ right \] this equation can be examples of the video used to fit step function:
That any one function How do we fit? The key to obtain the coefficient before the appropriate sine or cosine function .
2.1 introduces complex
Here we need to introduce complex that instead of sine or cosine function by complex functions. Since input and output are real domain function on the real axis, i.e. only one shaft, understood from this perspective, the real function is essentially one-dimensional domain, which does not help us to understand intuitively.
We know Euler's formula:
\ [^ {E} = IX \ COS X + I \ SiN X \]
Considering the conduction equation we need antipyretic cosine function, so we obtain a modification of the Euler's formula:
\ [2 \ cos (x) = e ^ {
ix} + e ^ {- ix} \] considering the problems heat conduction is about changes of time, so we will here \ (X \) is replaced with the time \ (T \) , the Fourier series is the essence of this complex exponential function:
\ [^ {iT} E \]
this function means, its output value will be at a rate of about 1 per unit rotation of the unit circle, this time It may be somewhat difficult to understand, in the last chapter will explain why the note, and now just need to know in this sense can be.
After transformation of the Euler equation, we get \ (cos (t) \) and \ (e ^ {it} \ ) relationship. We function broken down into a series of small and vectors, and these vectors are rotated in a small fixed integer frequency, by superposing the rotation vector, we can obtain an arbitrary curve, i.e. arbitrary function. But here is the rotation by the complex exponential function \ (e ^ {it} \ ) is described, and finally we only need modification or Euler's formula Euler's formula to convert it into a sine or cosine function can.
The rotation of each vector, we described by this formula:
\ [C_ {E} ^ {n-n-\ CDOT 2 \ PI IT} \]
where \ (2 \ pi \) means that a counterclockwise circle, and \ (2 \ \ pi it) coefficient before the rate is determined rotation, before the exponential function \ (C_N \) simultaneously determines the initial position and length of the arrow arrow starts to rotate, if multiplied by a representative of the selected complex exponential, e.g. \ (E {^ (\ PI /. 4) I} \) , it means that the arrow is rotated from the 45 ° position, if multiplied by a constant, changing the length of the arrow means:
These complex exponential functions, our aim is to describe arbitrary function (curve), as mentioned above, too: we need this function by itself, to a find a coefficient of each item:
\ [f ( t) = \ cdots + c _ {- 2} e ^ {- 2 \ cdot 2 \ pi it} + c _ {- 1} e ^ {- 1 \ cdot 2 \ pi it} + c_ {0} e ^ {0 \ cdot 2 \ pi it} +
c_ {1} e ^ {1 \ cdot 2 \ pi it} + c_ {2} e ^ {2 \ cdot 2 \ pi it} + \ cdots \] the simplest is to find the middle the constant term, which represents a center of gravity of the entire curve (in a time hypothesis \ (T \) for each discrete point in uniformity mass are equal, then this is really a center of gravity in the physical sense), mathematically understood this \ (C_0 \) is actually the integral of this curve function from 0 to 1:
\ [C_ {0} = \ {0} the int_. 1} ^ {F (T) dt \]
to understand the concept is simple, we will be converted into points and integral, in addition to \ (C_0 \) , each of the integral is 0, as in the second, these arrows are at a uniform speed about the origin of the circle of rotation of an integral multiple of .
Through this example, we found that in this infinite number, only the integral constant term is not zero. Therefore, we may be calculated by any one of this nature, on the basis of only the original function of a superior \ (E ^ {- n-\ CDOT 2 \ PI IT} \) , can be \ (C_N \) term becomes constant item, this way, \ (c_n \) of the integral term than the term to all zeros, while \ (c_n \) of the integral term is \ (c_n \) values, so that we get the calculation of \ (C_N \) method:
\ [n-C_ {} = \ {0} the int_. 1} ^ {F (T) E ^ {- n-\ CDOT 2 \ PI IT} dt \]
calculated integral to the computer can by iterative numerical solution can be obtained.
3 appreciated \ (e ^ {it} \ )
Finally, we note to explain why the article \ (e ^ {it} \ ) means that the output value will be at a rate of about 1 per unit rotation of the unit circle.
Exponential function \ (e ^ {x} \ ) important properties is that it is his own derivative. Let's speed and location (note that this is not the position displacement) approach to understanding this matter, we will time \ (t \) as a function of the input, we will find the real laws of motion of the shaft output of this function in very specific: the speed (the derivative value) of the output function of the output value of the position movement (function value) will always be equal . When we time \ (t \) in front of a real number multiplied by a coefficient, by the chain rule we know: After subsequent derivation of this real number will be multiplied to the function itself. This leads rate (derivative value) of the output function of the output value of the position movement (function value) always remains a fixed multiple .
With this understanding, we extend the problem to the complex plane. At time (T \) \ premultiplied by a \ (I \) was our function \ (E ^ {IT} \) . That have anything to do with the speed output function of the output value of the position of this movement?
We abandon the understanding of value, the \ (* i \) this calculation be understood as a behavior . We is defined as \ (I \) is:
\ [I ^ 2 = -1 \]
This equation is not seem intuitive, we get what shape:
\ [= I * I -1 \]
also not intuitive, continued:
\ [1 * i * i =
-1 \] this time we combined the complex plane to feel this equation:
We just mentioned, to \ (* i \) this calculation be understood as a behavior , we feel that through this chart is a look at this kind of behavior: the real number is continuously applied twice this behavior, it becomes 1 -1. That we through two \ (* i \) of this behavior, the real number 1 in the complex plane rotated 180 °, which means \ (* i \) this behavior is actually rotates counterclockwise 90 °, counterclockwise as to why, given the diet, mathematicians plural axis positive direction in the complex plane is defined as a, so a real number \ (* i \) a 90 ° counterclockwise rotation will fall position \ ((0, I) \) .
Learned \ (* i \) is rotated in a counter-clockwise 90 °, back \ (e ^ {it} \ ) function:
We found that the function of the output speed of movement of the output value of the position is always vertical! Always position (here can be understood as the position vector) vertical speed will result in what sport? Obviously, a circular motion .
\ (t = 0 \) when, \ (E ^ {IT} = 1 \) , which is our initial state through the above-mentioned, this formula means a rate of about 1 per unit rotation of the unit circle, when the \ (t = \ pi \) when rotated through 180 °, i.e. -1 if \ (t = 2 \ pi \ ) , then just turned one turn:
