51 lines
874 B
Typst
51 lines
874 B
Typst
#import "@preview/physica:0.9.6": *
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#import "lib.typ": ccases, fourier
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= problem 3
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== a)
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the truncated fourier series of a function looks like
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$
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f_N = #fourier(N: $N$)
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$
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note: i use $a_0/2$ with a $1/L$ in the definition of $a_0$.
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then to calculate the error, we take the difference with the actual function,
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which can be expressed as an infinite fourier series
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$
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e_N (x) & = f(x) - f_N (x) \
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& = [#fourier()] \
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& - [#fourier(N: $N$)] \
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& = sum_(n=N+1)^oo (a_n cos(n x) + b_n sin(n x))
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$
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it is the tail end of the infinite sum.
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parsevals identity states that
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$
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angle.l f, f angle.r =
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1/L integral_(-L)^L abs(f(x))^2 dd(x)
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= a_0^2/2 + sum_(n=1)^oo (a_n^2 + b_n^2)
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$
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== b)
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let $f(x) = e^(-x)$
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we first need to find the fourier coefficients
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$
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a_0 = integral_(-1)^1 e^(-x) dd(x)
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= [e^x]_(-1)^1 = e - 1/e
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$
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$
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a_n =
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$
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