Generally we can expand function in terms of Chebyshov polynomials f(x) = \sum_{n=0}^{\infty}{a_{n}T_{n}(x)} where a_{n} = \frac{2-\delta_{n0}}{\pi}\int_{0}^{\pi}{f(cos(t))cos(nt)dt} \delta_{nm} is Kronecker delta f(x) is polynomial here so we can calculate this integral without integration Suppose that you have function -2(1+x)^{5} and you want to calculate coefficient in front of T_{5}(x) where T_{n}(x) is Chebyshov T polynomial To expand function in terms of Chebyshov polynomials on the one side you can calculate it using integral \frac{(2-\delta_{n0})}{\pi}\int_{0}^{\pi}f(cos(t))cos(nt)dt , where n non-negative integer number but in the other side you can can calculate it without integral \delta_{nm} is Kronecker symbol So I suggest to expand function -2(1+x)^{5} in terms of Chebyshov polynomials without using integral -2(1+x)^{5} = a_{5}(16x^{5}-20x^{3}+5x)+a_{4}(8x^{4}-8x^{2}+1)+a_{3}(4x^{3}-3x)+a_{2}(2x^2-1)+a_{1}x+a_{0} but we look for coefficient in front of T_{5}(x) so -2 = 16a_{5} a_{5} = -1/8 2/pi * I= -1/8 I = pi/2*(-1/8) I = -pi/16 To prepare this integral for use of Chebyshov polynomials use substitution t = pi - x , and fact that integrand is even on interval symmetric around zero
It may look like integral from calculating coefficient of Fourier series expansion with cosine but it expands function in terms of Chebyshov polynomials instead of monomials like in Taylor expansion
@@cipherunity Name of this guy in original Cyrylic is Пафнутий Львович Чебышёв And vowel ё is read as yo Львович means that first name of his father was Lev If you want to read about it in original you can type Многочлены Чебышёва Here expressing functions in terms of Chebyshov polynomials can be useful In numerical analysis there is a topic about approximating functions using Chebyshov polynomials
Integrate (1 - cos x)^5 cos(5*x) dx from 0 to 2π = Integrate (1 - 5cos (x)+10cos^2(x)-10cos^3(x)+5cos^4(x)-5*cos^5(x)) cos(5*x) dx from 0 to 2π = Integrate (63/8-105/8*cos(x)+15/2*cos(2*x)-45/16*cos(3*x)+5/8*cos(4*x)-1/16*cos(5*x))*cos(5*x) dx from 0 to 2π =Integrate (63/8-105/8*cos(x)+15/2*cos(2*x)-45/16*cos(3*x)+5/8*cos(4*x))*cos(5*x) dx from 0 to 2π - 1/16 Integrate cos^2(5*x) dx from 0 to 2π =0-1/16*π = -π/16 Note: (i) Integrate cos (n*x) dx from 0 to 2π = 0 (ii) Integrate cos(m*x)*cos (n*x) dx from 0 to 2π =0 ,if m≠n (iii) Integrate cos(m*x)*cos (n*x) dx from 0 to 2π =π , if m=n
This integral is like tailor-made for the reciprocal beta function, it will directly get you the final result, only takes like 30 seconds.
I shall do that. Thanks for your advise.
Sir, it is done. Thanks again
Generally we can expand function in terms of Chebyshov polynomials
f(x) = \sum_{n=0}^{\infty}{a_{n}T_{n}(x)}
where
a_{n} = \frac{2-\delta_{n0}}{\pi}\int_{0}^{\pi}{f(cos(t))cos(nt)dt}
\delta_{nm} is Kronecker delta
f(x) is polynomial here so we can calculate this integral without integration
Suppose that you have function -2(1+x)^{5}
and you want to calculate coefficient in front of T_{5}(x)
where T_{n}(x) is Chebyshov T polynomial
To expand function in terms of Chebyshov polynomials on the one side you can calculate it using integral
\frac{(2-\delta_{n0})}{\pi}\int_{0}^{\pi}f(cos(t))cos(nt)dt , where n non-negative integer number
but in the other side you can can calculate it without integral
\delta_{nm} is Kronecker symbol
So I suggest to expand function -2(1+x)^{5}
in terms of Chebyshov polynomials without using integral
-2(1+x)^{5} = a_{5}(16x^{5}-20x^{3}+5x)+a_{4}(8x^{4}-8x^{2}+1)+a_{3}(4x^{3}-3x)+a_{2}(2x^2-1)+a_{1}x+a_{0}
but we look for coefficient in front of T_{5}(x) so
-2 = 16a_{5}
a_{5} = -1/8
2/pi * I= -1/8
I = pi/2*(-1/8)
I = -pi/16
To prepare this integral for use of Chebyshov polynomials
use substitution t = pi - x , and fact that integrand is even on interval symmetric around zero
It may look like integral from calculating coefficient of Fourier series expansion with cosine
but it expands function in terms of Chebyshov polynomials instead of monomials like in Taylor expansion
I need to read about Chebyshov polynomials
@@cipherunity Name of this guy in original Cyrylic is Пафнутий Львович Чебышёв
And vowel ё is read as yo
Львович means that first name of his father was Lev
If you want to read about it in original
you can type Многочлены Чебышёва
Here expressing functions in terms of Chebyshov polynomials can be useful
In numerical analysis there is a topic about approximating functions using Chebyshov polynomials
Integrate (1 - cos x)^5 cos(5*x) dx from 0 to 2π
= Integrate (1 - 5cos (x)+10cos^2(x)-10cos^3(x)+5cos^4(x)-5*cos^5(x)) cos(5*x) dx from 0 to 2π
= Integrate (63/8-105/8*cos(x)+15/2*cos(2*x)-45/16*cos(3*x)+5/8*cos(4*x)-1/16*cos(5*x))*cos(5*x) dx from 0 to 2π
=Integrate (63/8-105/8*cos(x)+15/2*cos(2*x)-45/16*cos(3*x)+5/8*cos(4*x))*cos(5*x) dx from 0 to 2π - 1/16 Integrate cos^2(5*x) dx from 0 to 2π
=0-1/16*π = -π/16
Note: (i) Integrate cos (n*x) dx from 0 to 2π = 0 (ii) Integrate cos(m*x)*cos (n*x) dx from 0 to 2π =0 ,if m≠n
(iii) Integrate cos(m*x)*cos (n*x) dx from 0 to 2π =π , if m=n
It is done. I just posted a new video using you solution
lol not sick but very long