Exponential Function Assignment | Online Assignment

This is a math writing project, which needs at least 4 pages. The requirements and template are in the following uploaded files. The topic of the writing project will be mainly about writing proofs for definitions. In the uploaded files, the requirements on 4th and 5th pages can be ignored.1 Introduction
In this essay, we discuss the exponential function. We will prove that:
• There is a real number:
e = 1+
1
1!
+
1
2!
+ · · ·
• There is a continues function exp : R ! R such that exp(1) = e, and
exp(x + y) = exp(x) · exp(y) for all x, y 2 R.
• We can extend exp to a continuous function C ! C.
We may also prove that:
• For all x, y 2 R, exp(x) < exp(y) if and only if x < y.
• For all z 2 C, exp(z) 6= 0.
• The function exp is equal to its own derivative.
• e⇡i + 1 = 0.
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2 Instructions
2.1 Bare minimum
If you want to pass, you will have to do the following:
• Submit a typed essay by Sunday August 2nd.
• Don’t cheat.
• The essay should, at the very least, prove that there exists a function:
exp : Q ! R exp(1) = lim
n!1
Xn
k=0
1
k!
If you do these things and nothing more, the essay you submit is your
own work, and there are no fatal errors, then you will not fail, but you will
probably not get much higher than a C.
A fatal error is a mistake that makes the entire proof unsalvageable.
• If you forget to invoke a lemma, or you forget to consider a special
case, then that is probably not a fatal error. For example, if you prove
something for all ” > 0 and you forget that “2 > ” if ” > 1, that is a
mild error.
• If you write −i < i (where i2 = −1), that is a fatal error – I don’t care
what you think you were doing, but is not an ordered field and you
should know that.
2.2 If you need a B−
• Make your essay readable to humans – include an introduction, a conclusion,
and in between proofs, explain what you’re doing and why
you’re doing it.
• Prove there exists a continuous function exp : R ! R.
• Don’t cheat.
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tnssnsososonsseesnsnessressosnn
2.3 If you want an A− or higher…
• Do everything needed for a B−.
• Prove something about ex which was not assigned as a homework problem.
Possible things you can include:
– Prove there exists a continuous function C ! C that extends ex.
– Prove that ex is di↵erentiable, and that it is equal to its own
derivative.
– Prove cos2 z + sin2 z = 1 for all z 2 C. Deduce Euler’s formula.
– Prove that ex is a homomorphism – that is, ex+y = ex · ey.
2.4 If you’re aiming for an A or an A+.
• There should be almost no mistakes or typos – the number I will tolerate
will depend on the length of the overall paper.
• Either:
– Include multiple items from the A− list.
– Discuss a di↵erent advanced topic (that I agree to in advance).
⇤ You can talk about something that relates the exponential to
your interests.
⇤ You can do the p-adics project (3).
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3 p-adics
3.1 Set-up
Let p 2 N be a prime number. We will define a new absolute value on Q
that is determined by p.
• Let x 2 Q and assume that x > 0. Then there exist unique r, s 2 N
such that x = r
s and r, s are relatively prime (i.e. they have no factors
in common).
• Since r, s are relatively prime, exactly one of the following is true:
– p divides r and p does not divide s.
– p divides s but p does not divide r.
– p does not divide r or s.
In other words, it is not possible for p to divide r and s, since we are
assuming they have no factors in common.
• Since r, s are unique for each x 2 Q, we can use themto define functions
on Q. We define |x| = 1 if p does not divide r or s.
• Let m 2 N be a nonzero natural number and suppose p divides m.
By the fundamental theorem of arithmetic, there is a positive integer k
such that pk divides m and pk+1 does not divide m. We define |m| = 1
pk .
• If p divides r and p does not divide s, we define |r
s || = |r|.
• If p divides s and does not divide r, we define |r
s | = 1
|s|
.
• Finally, we define |0| = 0 and |−x| = |x| for all x < 0.
3.2 Problems
Do the following:
1. Prove that |xy|p = |x|p|y|p for all x, y 2 Q.
2. Let x 2 Z. Prove that |x|p  1.
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3. Let x, y 2 Z, and assume |x|p < |y|p. Prove that |x ± y|p = |y|p.
4. Let x, y 2 Z and assume |x| = |y|. Prove that |x ± y|p  |x|p = |y|p.
5. Prove that | · |p is an absolute value on Q.
6. Let {an} ⇢ Q be a sequence, and define a new sequence s0 = a0,
sn+1 = sn + an+1.
Prove that {|sn|p}n2N is Cauchy if and only if lim |an|p = 0.
7. Let an = n!. Prove that lim an = 0 with respect to ||p. That is, prove:
(8″ > 0)(9N 2 N)(8n > N)(|n!|p < “)
8. Let a 2 N. Prove that lim|an|p = 0 if and only if p divides a.
9. Let x 2 Q and let ex
n be the usual sequence. Prove that ex
n is Cauchy
if and only if p divides the numerator of x and p does not divide the
denominator.
3.3 Extra things
If (and only if) you do the previous exercises and want to do more to guarantee
an A+, also do some or all of the following.
1. Let x 2 Q. Prove that |xp| = 1 for all but finitely many primes p.
2. Let x 2 Q. Prove that:
|x|R ·
Y
p |x|p
Remark: While there are infinitely many primes, the product on the
right hand side is really a finite product, since only finitely many terms
are not equal to 1.
3. Let x 2 Q. Prove that x 2 Z if and only if |x|p  1 for all primes p.
4. What geometric property from lecture does Z have as a subset of the
metric space Q, with respect to ||p for all p, that it does not have with
respect to the usual absolute value?
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