Finished lab1
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2 changed files with 178 additions and 13 deletions
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@ -1,5 +1,6 @@
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from sys import argv
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from math import pi
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from math import e
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function_formulas = [
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# 1. Square wave
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@ -29,6 +30,18 @@ function_formulas = [
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if n > 0 else
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(4 / (pi ** 2 * n ** 2) if n % 2 == 0 else -24j / (pi ** 3 * n ** 3))
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)
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},
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# 5. Complex func
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{
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'an': lambda n: 0,
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'bn': lambda n: 0,
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'cn': lambda n: 0 if n == 0 else (
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complex(-pi * n / 4, -1 - pi * n / 4) * e ** (1j * pi * n / 4)
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+ (1j + 1) * e ** (-1j * pi * n / 4)
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+ (1j - 1) * e ** (-3j * pi * n / 4)
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+ (-1j - 1) * e ** (-5j * pi * n / 4)
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+ complex(1 + pi * n / 4, pi * n / 4) * e ** (-7j * pi * n / 4)
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) / (pi ** 2 * n ** 2 / 2)
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}
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]
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@ -87,15 +87,15 @@ $ b_n &= 1/(T/2) integral^3_1 f(t)sin(omega_n t)upright(d)t = integral^3_1 f(t)s
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2/(pi n)\, & n = 2k - 1
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), k in ZZ $
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$ c_n &= 1/T integral^3_1 f(t)e^(i omega_n t) = 1/2 integral^3_1 f(t)e^(i pi n t) = 1/2 (integral^2_1 e^(i pi n t) + integral^3_2 2e^(i pi n t))upright(d)t \
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&= 1/2 (lr(-(i e^(i pi n t))/(pi n)|)^2_1 - lr((2i e^(i pi n t))/(pi n)|)^3_2) = -(i e^(2i pi n))/(2pi n) + (i e^(i pi n))/(2pi n) - (2i e^(3i pi n))/(2pi n) + (2i e^(2i pi n))/(2pi n) \
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&= (i e^(2i pi n))/(2pi n) - (i e^(i pi n))/(2pi n) = i/(2pi n) - (i(-1)^n)/(2pi n) = (i(1 - (-1)^n))/(2pi n) \
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$ c_n &= 1/T integral^3_1 f(t)e^(-i omega_n t) = 1/2 integral^3_1 f(t)e^(-i pi n t) = 1/2 (integral^2_1 e^(-i pi n t) + integral^3_2 2e^-(i pi n t))upright(d)t \
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&= 1/2 (lr(-(i e^(-i pi n t))/(pi n)|)^2_1 - lr((2i e^(-i pi n t))/(pi n)|)^3_2) = -(i e^(-2i pi n))/(2pi n) + (i e^(-i pi n))/(2pi n) - (2i e^(-3i pi n))/(2pi n) + (2i e^(-2i pi n))/(2pi n) \
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&= (i e^(-2i pi n))/(2pi n) - (i e^(-i pi n))/(2pi n) = i/(2pi n) - (i(-1)^n)/(2pi n) = (i(1 - (-1)^n))/(2pi n) \
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&= cases(
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0\, & n = 2k,
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i/(pi n)\, & n = 2k - 1
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), k in ZZ $
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$ a_0 &= c_0 = 3, a_n = 0, \
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$ a_0 &= 3, a_n = 0, c_0 = a_0\/2 = 1.5 \
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b_n &= cases(
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0\, & n = 2k,
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2/(pi n)\, & n = 2k - 1
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@ -554,7 +554,7 @@ $ f(t) = cases(
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Период функции $T = 4$, $omega_n = (pi n)/2$. Рассмотрим функцию на промежутке $[-2; 2]$ и найдём коэффициенты $a_0, a_n, b_n, c_n$:
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$ a_n &= 1/(T/2) integral^2_(-2) f(t) upright(d)t \
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$ a_0 &= 1/(T/2) integral^2_(-2) f(t) upright(d)t \
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&= ... \
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&= -2/3 $
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@ -597,13 +597,13 @@ function_formulas = [
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'cn': lambda n: -1/3 if n == 0 else (
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(4 / (pi ** 2 * n ** 2) if n % 2 == 0 else 24j / (pi ** 3 * n ** 3))
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if n > 0 else
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(4 / pi ** 2 * n ** 2) if n %2 == 0 else -24j / (pi ** 3 * n ** 3))
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(4 / pi ** 2 * n ** 2) if n % 2 == 0 else -24j / (pi ** 3 * n ** 3))
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)
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}
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...
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]
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...
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```, caption: "Дополнение программы нечётной функцией")
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```, caption: "Дополнение программы четвёртой функцией")
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#figure(```
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$ python fourier.py 4 3
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@ -754,7 +754,7 @@ $ upright("Re")f(t) = cases(
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} else if tt >= t1 and tt < t2 {
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return 1
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} else if tt >= t2 and tt < t3 {
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return 4-t
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return 4-tt
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} else {
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return -1
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}
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@ -767,8 +767,160 @@ $ upright("Re")f(t) = cases(
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Период функции $T = 8$, $omega_n = (pi n)/4$. Рассмотрим функцию на промежутке $[-1; 7]$ и найдём коэффициент $c_n$:
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$ c_n &= 1/T integral^7_(-1) f(t)e^(i omega_n t) upright(d)t = \
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&= ... \
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&= (((omega_n-1)sin(omega_n) + omega_n cos(omega_n))(1+i e^(2i omega_n) -i e^(4i omega_n) -i e^(6i omega_n)))/(4omega_n^2) \
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&= ((((pi n)/4-1)sin((pi n)/4) + (pi n)/4 cos((pi n)/4))(1+i e^(i (pi n)/2) -i e^(i pi n) -i e^(i (3pi n)/2)))/((pi^2 n^2)/4) $
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$ c_n &= 1/T integral^7_(-1) f(t)e^(-i omega_n t) upright(d)t \
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&= 1/8(integral^1_(-1) (1+i t)e^(-i omega_n t) upright(d)t + integral^3_1 (2-t+i)upright(d)t + integral^5_3 (-1+i(4-t))upright(d)t + integral^7_5 (-6+t-i)upright(d)t) \
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&= 1/8(integral^1_(-1) e^(-i omega_n t)upright(d)t + 2integral^3_1 e^(-i omega_n t)upright(d)t + i integral^3_1 e^(-i omega_n t)upright(d)t - integral^5_3 e^(-i omega_n t)upright(d)t + 4i integral^5_3 e^(-i omega_n t)upright(d)t \
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&- 6integral^7_5 e^(-i omega_n t)upright(d)t - i integral^7_5 e^(-i omega_n t)upright(d)t + i integral^1_(-1) t e^(-i omega_n t)upright(d)t - integral^3_1 t e^(-i omega_n t)upright(d)t - i integral^5_3 t e^(-i omega_n t)upright(d)t + integral^7_5 t e^(-i omega_n t)upright(d)t) \
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&= lr((i e^(-i omega_n t))/(8omega_n)|)^1_(-1) + lr((2i e^(-i omega_n t))/(8omega_n)|)^3_1 - lr((e^(-i omega_n t))/(8omega_n)|)^3_1 - lr((i e^(-i omega_n t))/(8omega_n)|)^5_3 - lr((4 e^(-i omega_n t))/(8omega_n)|)^5_3 - lr((6i e^(-i omega_n t))/(8omega_n)|)^7_5 + lr((e^(-i omega_n t))/(8omega_n)|)^7_5 \
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&+ lr((e^(-i omega_n t)(i-omega_n t))/(8omega_n^2)|)^1_(-1) - lr((e^(-i omega_n t)(1+i omega_n t))/(8omega_n^2)|)^3_1 - lr((e^(-i omega_n t)(i-omega_n t))/(8omega_n^2)|)^5_3 + lr((e^(-i omega_n t)(1+i omega_n t))/(8omega_n^2)|)^7_5 \
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&= ((-i-omega_n-i omega_n)e^(i omega_n) + (i+1)e^(-i omega_n) + (i-1)e^(-3i omega_n) + (-i-1)e^(-5i omega_n) + (omega_n + 1 + i omega_n)e^(-7i omega_n))/(8omega_n^2) \
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&= ((-i-(pi n)/4-i (pi n)/4)e^(i (pi n)/4) + (i+1)e^(-i (pi n)/4) + (i-1)e^(-3i (pi n)/4) + (-i-1)e^(-5i (pi n)/4) + ((pi n)/4 + 1 + i (pi n)/4)e^(-7i (pi n)/4))/((pi^2 n^2)/2) $
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$ c_0 &= 1/T integral^7_(-1)f(t)upright(d)t \
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&= 1/8(integral^1_(-1) (1+i t) upright(d)t + integral^3_1 (2-t+i) upright(d)t + integral^5_3 (-1+i(4-t)) upright(d)t + integral^7_5 (-6+t-i) upright(d)t) \
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&= 1/8(lr((t+(i t^2)/2)|)^1_(-1) + lr((2t+i t-t^2/2)|)^3_1 + lr((-t+4i t-(i t^2)/2)|)^5_3 + lr((-6t+t^2/2-i t)|)^7_5) \
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&= 1/8(1+i/2+1-i/2+6+3i-9/2-2-i+1/2-5+20i-(25i)/2+3-12i+(9i)/2-42+49/2 \
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&-7i+30-25/2+5i) = 0 $
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#pagebreak()
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Дополним программу @prog новой комплексной функцией:
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#figure(```python
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...
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function_formulas = [
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...
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# 5. Complex func
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{
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'an': lambda n: 0,
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'bn': lambda n: 0,
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'cn': lambda n: 0 if n == 0 else (
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complex(-pi * n / 4, -1 - pi * n / 4) * e ** (1j * pi * n / 4)
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+ (1j + 1) * e ** (-1j * pi * n / 4)
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+ (1j - 1) * e ** (-3j * pi * n / 4)
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+ (-1j - 1) * e ** (-5j * pi * n / 4)
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+ complex(1 + pi * n / 4, pi * n / 4) * e ** (-7j * pi * n / 4)
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) / (pi ** 2 * n ** 2 / 2)
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}
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...
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]
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...
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```, caption: "Дополнение программы комплексной функцией")
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#figure(```
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$ python fourier.py 5 3
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an values: ['0: 0', '1: 0', '2: 0', '3: 0']
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bn values: ['0: 0', '1: 0', '2: 0', '3: 0']
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cn values: ['-3: -0.12737+3.9996e-17j', '-2: 1.1249e-17+1.1249e-17j', '-1: 1.1249e-17+2.2498e-17j', '0: 0', '1: 1.1463+1.3499e-16j', '2: -6.7493e-17-6.7493e-17j', '3: 3.9996e-17+1.9998e-17j']
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```, caption: "Ввывод программы для копмлексной функции")
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Из-за погрешностей чисел с плавающей точкой (float), программа выводит очень маленькое число ($~10^(-17)$) в некоторых случаях. В таких ситуациях мы будем воспринимать коэффициент $c_n$ равным нулю.
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$ G_3(t) = -0.12737e^((-3i pi t)/4) + 1.1463e^((i pi t)/4) $
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С помощью программы из @prog, подсчитаем также коэффициенты для $N = 10$, после чего построим график $G_N (t)$ для значений $N = 1, 2, 3, 10$ и сравним их с $f(t)$:
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#pagebreak()
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#figure(
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canvas(length: 1.75cm, {
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plot.plot(size: (8, 4),
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x-tick-step: 1, y-tick-step: 1,
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x-min: -10, x-max: 10,
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y-min: -3, y-max: 3,
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x-grid: true, y-grid: true,
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x-label: [$t$], y-label: [Re$G_N (t)$],
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{
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plot.add(domain: (-9.99, 9.99), t => {
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let (t0, t1, t2, t3, t4) = (-1, 1, 3, 5, 7)
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if t < t0 { t = -t + t0 + 1 }
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let tt = calc.rem(t - t0, t4 - t0) + t0
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if tt >= t0 and tt < t1 {
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return 1
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} else if tt >= t1 and tt < t2 {
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return 2 - tt
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} else if tt >= t2 and tt < t3 {
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return -1
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} else {
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return -6 + tt
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}
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}, line: "linear", samples: 1000, label: [Re$f(t)$#v(1em)])
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plot.add(domain: (-9.99, 9.99), t => {
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1.1463*calc.cos(calc.pi*t/4)
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}, line: "spline", samples: 100, label: [Re$G_1(t)$#v(1em)])
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plot.add(domain: (-9.99, 9.99), t => {
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-0.12737*calc.cos(3*calc.pi*t/4) + 1.1463*calc.cos(calc.pi*t/4)
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}, line: "spline", samples: 100, label: [Re$G_2(t)$#v(1em)])
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plot.add(domain: (-9.99, 9.99), t => {
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-0.12737*calc.cos(3*calc.pi*t/4) + 1.1463*calc.cos(calc.pi*t/4)
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}, line: "spline", samples: 100, label: [Re$G_3(t)$#v(1em)], style: (stroke: purple))
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plot.add(domain: (-9.99, 9.99), t => {
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0.023394*calc.cos(7*calc.pi*t/4) - 0.12737*calc.cos(3*calc.pi*t/4) + 1.1463*calc.cos(calc.pi*t/4) - 0.045853*calc.cos(5*calc.pi*t/4) + 0.014152*calc.cos(9*calc.pi*t/4)
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}, line: "spline", samples: 100, label: [Re$G_10(t)$#v(1em)])
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}
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)
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}), caption: [Графики частичных сумм Фурье комплексной функции $G_N (t)$ и $f(t)$ (Вещественная плоскость)]
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)
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#figure(
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canvas(length: 1.75cm, {
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plot.plot(size: (8, 4),
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x-tick-step: 1, y-tick-step: 1,
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x-min: -10, x-max: 10,
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y-min: -3, y-max: 3,
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x-grid: true, y-grid: true,
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x-label: [$t$], y-label: [Im$G_N (t)$],
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{
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plot.add(domain: (-9.99, 9.99), t => {
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let (t0, t1, t2, t3, t4) = (-1, 1, 3, 5, 7)
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if t < t0 { t = -t + t0 - 3 }
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let tt = calc.rem(t - t0, t4 - t0) + t0
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if tt >= t0 and tt < t1 {
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return tt
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} else if tt >= t1 and tt < t2 {
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return 1
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} else if tt >= t2 and tt < t3 {
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return 4-tt
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} else {
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return -1
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}
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}, line: "linear", samples: 1000, label: [Im$f(t)$#v(1em)], style: (stroke: yellow))
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plot.add(domain: (-9.99, 9.99), t => {
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1.1463*calc.sin(calc.pi*t/4)
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}, line: "spline", samples: 100, label: [Im$G_1(t)$#v(1em)])
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plot.add(domain: (-9.99, 9.99), t => {
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0.12737*calc.sin(3*calc.pi*t/4) + 1.1463*calc.sin(calc.pi*t/4)
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}, line: "spline", samples: 100, label: [Im$G_2(t)$#v(1em)])
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plot.add(domain: (-9.99, 9.99), t => {
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0.12737*calc.sin(3*calc.pi*t/4) + 1.1463*calc.sin(calc.pi*t/4)
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}, line: "spline", samples: 100, label: [Im$G_3(t)$#v(1em)], style: (stroke: purple))
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plot.add(domain: (-9.99, 9.99), t => {
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-0.023394*calc.sin(7*calc.pi*t/4) + 0.12737*calc.sin(3*calc.pi*t/4) + 1.1463*calc.sin(calc.pi*t/4) - 0.045853*calc.sin(5*calc.pi*t/4) + 0.014152*calc.sin(9*calc.pi*t/4)
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}, line: "spline", samples: 100, label: [Im$G_10(t)$#v(1em)])
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}
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)
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}), caption: [Графики частичных сумм Фурье комплексной функции $G_N (t)$ и $f(t)$ (Комплексаня плоскость)]
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)
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Из графиков видно, что сумма Фурье $G_N (t)$ сходится к исходной функции $f(t)$ при $N -> infinity$ - коэффициент $c_n$ подсчитан верно.
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#pagebreak()
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Проверим выполнение равенства Парсеваля:
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$ 1/(T/2) integral^7_(-1) (f(t))^2 upright(d)t = sum^infinity_(n=-infinity) |c_n|^2 $
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$ 1/(T/2) integral^7_(-1) (f(t))^2 upright(d)t &= 1/4(integral^1_(-1) (1+i t)^2 upright(d)t + integral^3_1 (2-t+i)^2 upright(d)t integral^5_3 (-1+i(4-t))^2 upright(d)t + integral^7_5 (-6+t-i)^2 upright(d)t) \
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&= 1/4(lr((t+i t^2-t^3/3)|)^1_(-1) + lr((3t+4i t-2t^2-i t^2+t^3/3)|)^3_1 \
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&+ lr((-15t-8i t+i t^2+4t^2-t^3/3)|)^5_3 + lr((t^3/3-6t^2-i t^2+35t+12i t)|)^7_5)\
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&= 1/4(1+i-1/3+1-i-1/3+9+12i-18-9i+9-3-4i+2+i-1/3\
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&-75-40i+25i+100-125/3+45+24i-9i-36+9+343/3-294-49i\
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&+245+84i-125/3+150+25i-175-60i) = 0 $
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$ sum^infinity_(n=-infinity) |c_n|^2 &= sum^infinity_(n=-infinity) lr(|cases(
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(8sqrt(2))/(pi^2 n^2)\, & n = 8k-7,
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-(8sqrt(2))/(pi^2 n^2)\, & n = 8k-3
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)|)^2, k in ZZ = sum^infinity_(n=-infinity) ((8sqrt(2))/(pi^2 n^2))^2, n = 4k-3, k in ZZ \
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&= 128/pi^4 sum^infinity_(k=-infinity) 1/(4k-3)^2 = 128/pi^4 dot pi^4/96 = 4/3 $
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Левая и правая часть не равны, поскольку функция $f(t)$ не соответствует условию Дирихле.
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