115 lines
4.5 KiB
Typst
115 lines
4.5 KiB
Typst
#import "@preview/physica:0.9.6": *
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= problem 1
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recall the definition of a periodic function $f$ for a $p > 0$
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$
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f(x + p) = f(x) quad forall quad x in RR.
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$
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the smallest such $p$ is called the fundamental period of $f$.
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== a)
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"every periodic function has a fundamental period" is a false statement.
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examine $f(x) = 1,$ which is periodic since given a $p = 1$, then $f(x + 1)
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= f(x) = 1$ for all $x in RR$. it is trivially periodic. however, there is no
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smallest $p$ for which this holds:
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given a $delta > 0$ there is always a $hat(delta) > 0$ for which $hat(delta)
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< delta$ holds. this is a property of the real number line, thus there is no
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fundamental period for this periodic function.
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== b)
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+ "$"Per"_p := {f: RR -> RR | f #[is $p$-periodic]}$ is a vector space" is
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a true statement since all periodic functions that are added together or
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scaled by some scalar are still periodic. this comes from the linear property
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of periodic functions.\
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we can prove this by taking two periodic functions $f$ and $g$ and seeing if
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their linear combination is an element of the space
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$
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h(t) & := a dot f(t) + b dot g(t) \
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& = a dot f(t + p_f) + b dot g(t + p_g) \
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& = a dot f(t + p_f dot p_g) + b dot g(t + p_g dot p_f) \
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& = h(t + p_f dot p_g) = h(t + p_h)
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$
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thus the linear combination $h(t)$ must be periodic itself and therefore an
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element of $"Per"_p$. this proves scalar multiplication and vector addition
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axioms, rest is trivial.
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+ "let $phi.alt : RR -> RR => phi.alt compose f$ is $p$-periodic" is true, since
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it 'captures' the input of the function $phi.alt$ such that it becomes
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periodic itself, always reiterating over the same values. \
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$
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(phi.alt compose f)(t) & = phi.alt(f(t)) \
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& = phi.alt(f(t + p)) \
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& = (phi compose f)(t + p)
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$
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+ "let $n in NN, a in RR => f(x + a), f(n x), f(x slash n)$ are $p$-periodic" is
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false, since scaling the input parameter changes the period. \
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$
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(1) &quad f(x + a) = f((x + p) + a) = f((x + a) + p)) #underline[ok] \
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(2) &quad f(n x) = f(n (x + p)) = f(n x + n p) = f(n x + p) #underline[ok] \
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(3) &quad f(x slash n) = f((x + p) slash n) = f(x slash n + p slash n) #underline[not ok]
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$
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(2) works because $n in NN$, such that the period repeats $n$ times. (3)
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doesn't work because dividing by a natural number causes the period to
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contract and thus isn't $p$-periodic anymore.
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+ "the absolute difference between two periods $p$ and $p'$ is also a period of
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$f$" is true, since it just means that the periods occur periodically. \
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$
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f(x) & = f(x + p') = f(x - p') \
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& = f((x + p) - p') = f(x + (p - p'))
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$
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given that $p > p'$.
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+ "let $a, b in RR => integral_a^(a + p) f(x) dd(x) = integral_b^(b + p) f(x) dd(x)$"
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is true, since it says that integrating over the period is the same regardless
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of where you start integrating from. \
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we can choose a $c in RR$ such that
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$
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integral_a^(a + p) f(x) dd(x)
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& = integral_a^c f(x) dd(x) + integral_c^(a + p) f(x) dd(x) \
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& = integral_c^(a + p) f(x) dd(x) + integral_a^c f(x) dd(x)
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= integral_b^(b + p) f(x) dd(x)
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$
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because $c$ is a midpoint chosen to displace the integral sum such that the
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startpoints of the integration are the same in terms of the period of the
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function $f$.
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+ "if $f$ is differentiable, $f'$ is also $p$-periodic" is true \
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$
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f(x) = f(x + p) ==> f'(x) = f'(x + p)
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$
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== c)
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- $f(x) = cos(2x + 3)$ has a fundamental period $p = pi$ since the $+ 3$ doesn't
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affect the period and then the usual period of $2 pi$ is halved by the
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coefficient in-front of $x$.
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- $f(x) = pi sin(3/2 pi x)$ has a fundamental period $p = 4 slash 3$ since the
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$pi$ in-front of the $sin$-expression only affects the amplitude and $(2 pi)
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/ ((3 slash 2) pi) = 4 slash 3$.
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- $f(x) = cos(pi/(m+1) x) + sin(pi/(n-1) x)$ for $m in ZZ \\ {-1}, n in ZZ \\ {1}$
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can be broken into two functions that can be analyzed separately first.
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- $g(x) := cos(pi/(m + 1) x)$ has a fundamental period $p_g = 2(m + 1)$.
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- $h(x) := sin(pi/(n + 1) x)$ has a fundamental period $f_h = 2(n + 1)$.
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- we can combine these two to obtain $f(x) = g(x) + h(x)$.
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to find the fundamental period, we can draw inspiration from number theory to
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see that the combined period must be
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$
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gcd(p_g, f_H) & = gcd(2(m + 1), 2(n + 1)) \
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& = gcd(m + 1, n + 1) \
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& <= (m + 1)(n + 1) \
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& = m n + m + n + 1
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$
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but we cannot shorten this further using $p$-period algebra, so this must be
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the fundamental period for this wave.
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