#import "@preview/physica:0.9.6": *

= problem 1

recall the definition of a periodic function $f$ for a $p > 0$
$
  f(x + p) = f(x) quad forall quad x in RR.
$
the smallest such $p$ is called the fundamental period of $f$.

== a)

"every periodic function has a fundamental period" is a false statement.

examine $f(x) = 1,$ which is periodic since given a $p = 1$, then $f(x + 1)
= f(x) = 1$ for all $x in RR$. it is trivially periodic. however, there is no
smallest $p$ for which this holds:

given a $delta > 0$ there is always a $hat(delta) > 0$ for which $hat(delta)
< delta$ holds. this is a property of the real number line, thus there is no
fundamental period for this periodic function.

== b)

+ "$"Per"_p := {f: RR -> RR | f #[is $p$-periodic]}$ is a vector space" is
  a true statement since all periodic functions that are added together or
  scaled by some scalar are still periodic. this comes from the linear property
  of periodic functions.\
  we can prove this by taking two periodic functions $f$ and $g$ and seeing if
  their linear combination is an element of the space
  $
    h(t) & := a dot f(t) + b dot g(t) \
         & = a dot f(t + p_f) + b dot g(t + p_g) \
         & = a dot f(t + p_f dot p_g) + b dot g(t + p_g dot p_f) \
         & = h(t + p_f dot p_g) = h(t + p_h)
  $
  thus the linear combination $h(t)$ must be periodic itself and therefore an
  element of $"Per"_p$. this proves scalar multiplication and vector addition
  axioms, rest is trivial.

+ "let $phi.alt : RR -> RR => phi.alt compose f$ is $p$-periodic" is true, since
  it 'captures' the input of the function $phi.alt$ such that it becomes
  periodic itself, always reiterating over the same values. \
  $
    (phi.alt compose f)(t) & = phi.alt(f(t)) \
                           & = phi.alt(f(t + p)) \
                           & = (phi compose f)(t + p)
  $

+ "let $n in NN, a in RR => f(x + a), f(n x), f(x slash n)$ are $p$-periodic" is
  false, since scaling the input parameter changes the period. \
  $
    (1) &quad f(x + a) = f((x + p) + a) = f((x + a) + p)) #underline[ok] \
    (2) &quad f(n x) = f(n (x + p)) = f(n x + n p) = f(n x + p) #underline[ok] \
    (3) &quad f(x slash n) = f((x + p) slash n) = f(x slash n + p slash n) #underline[not ok]
  $
  (2) works because $n in NN$, such that the period repeats $n$ times. (3)
  doesn't work because dividing by a natural number causes the period to
  contract and thus isn't $p$-periodic anymore.

+ "the absolute difference between two periods $p$ and $p'$ is also a period of
  $f$" is true, since it just means that the periods occur periodically. \
  $
    f(x) & = f(x + p') = f(x - p') \
         & = f((x + p) - p') = f(x + (p - p'))
  $
  given that $p > p'$.

+ "let $a, b in RR => integral_a^(a + p) f(x) dd(x) = integral_b^(b + p) f(x) dd(x)$"
  is true, since it says that integrating over the period is the same regardless
  of where you start integrating from. \

  we can choose a $c in RR$ such that
  $
    integral_a^(a + p) f(x) dd(x)
    & = integral_a^c f(x) dd(x) + integral_c^(a + p) f(x) dd(x) \
    & = integral_c^(a + p) f(x) dd(x) + integral_a^c f(x) dd(x)
    = integral_b^(b + p) f(x) dd(x)
  $
  because $c$ is a midpoint chosen to displace the integral sum such that the
  startpoints of the integration are the same in terms of the period of the
  function $f$.

+ "if $f$ is differentiable, $f'$ is also $p$-periodic" is true \
  $
    f(x) = f(x + p) ==> f'(x) = f'(x + p)
  $

== c)

- $f(x) = cos(2x + 3)$ has a fundamental period $p = pi$ since the $+ 3$ doesn't
  affect the period and then the usual period of $2 pi$ is halved by the
  coefficient in-front of $x$.

- $f(x) = pi sin(3/2 pi x)$ has a fundamental period $p = 4 slash 3$ since the
  $pi$ in-front of the $sin$-expression only affects the amplitude and $(2 pi)
  / ((3 slash 2) pi) = 4 slash 3$.

- $f(x) = cos(pi/(m+1) x) + sin(pi/(n-1) x)$ for $m in ZZ \\ {-1}, n in ZZ \\ {1}$
  can be broken into two functions that can be analyzed separately first.
  - $g(x) := cos(pi/(m + 1) x)$ has a fundamental period $p_g = 2(m + 1)$.
  - $h(x) := sin(pi/(n + 1) x)$ has a fundamental period $f_h = 2(n + 1)$.
  - we can combine these two to obtain $f(x) = g(x) + h(x)$.
  to find the fundamental period, we can draw inspiration from number theory to
  see that the combined period must be
  $
    gcd(p_g, f_H) & = gcd(2(m + 1), 2(n + 1)) \
                  & = gcd(m + 1, n + 1) \
                  & <= (m + 1)(n + 1) \
                  & = m n + m + n + 1
  $
  but we cannot shorten this further using $p$-period algebra, so this must be
  the fundamental period for this wave.