Let $f$ be analytic in the disc $|z| < 1$ and assume that $|f(z)| \le \dfrac{1}{1-|z|}$.
Show that $|{f}^{(n)}(0)| ≤ e(n + 1)!$.
Any ideas on how to bound $\max|f(z)|$ in the disc?
Isn't $2^0$ the first element and $2^4$ the fifth operation which results in the identity making ${\Bbb{Z}}_{5}$ order 5? I guess this is what the proof is referring to when it claims for x in ${Z}_{d}$ $x^d = 1 $ where d is a prime power.
I am looking at this proof and I am stuck on the logic that $a^{p}$ = 1. For example, consider the group under multiplication without zero, ${Z}_{5}$, wouldn't 2^4 = 1 imply that the order is 4 not 5? We know that if G is a finite abelian group, G is isomorphic to a direct product...
Consider the function $f(z) = e^{1/z}$,
Show that for any complex number ${w}_{0} \ne 0$ and any δ > 0, there exists ${z}_{0} ∈ C$
such that $ 0 < |{z}_{0}| < δ$ and $f({z}_{0}) = {w}_{0}$
I really don't know where to begin on this.
A mistake.
It's just $2017^{2}$ mod(100) which you can find with a calculator, but if you did not have a calculator could we transform $2017^{2}$ mod(100) further.
I don't know anything about it. We only went of Fermat's and Euler's generalization of ax$\equiv$ b mod(m). The class is a Ring Theory class so we aren't really diving that deeply into the number theory.
What are the last two digits of ${2017}^{82}$? We have only learned Fermat's little theorem and Euler's generalization so I am not sure how they apply?
Compute the remainder of 2^(2^17) + 1 when divided by 19. The book says to first compute 2^17 mod 18 but I don’t understand why we go to mod 18. Advice would be appreciated
Let f(z) = $e^{e^{z}}$ . Find Re(f) and Im(f).
I don't know how to deal with the exponential within an exponential. Does anybody know how to deal with this?
The question is, how can we prove $\lim_{{n}\to{n+1}}\int_{n}^{n+1}cos(2x^2)/2 \,dx = 0$? The best I have been able to come up with is the fact that $\lim_{{n}\to{n+1}}\ \int_{0}^{n+1}cos(2x^2)/2 \,dx - \int_{0}^{n}cos(2x^2)/2 \,dx$ = $\int_{n}^{n+1}cos(2x^2)/2 \,dx$ and each function is...