1
vote
2answers
49 views

Proving that the line integral $\int_{\gamma_{2}} e^{ix^2}\:\mathrm{d}x$ tends to zero

Let $f(z) = e^{iz^2}$ and $\gamma_2 = \{ z : z = Re^{i\theta}, 0 \leq \theta \leq \frac{\pi}{4} \} $. All the sources I have found online, says that the line integral $$ \left| \int_{\gamma_2} ...
1
vote
0answers
30 views

Performance estimation of shellSort

I'm trying to make a performance estimation for shell-sort algorithm. And I fail in it. My formula: equals to where dz is outer while-loop, dy is middle for-loop, and dx is inner for-loop ...
1
vote
1answer
37 views

Some mean value limiting result

Let $\phi$ be continuous in a neighborhood of $0\in\mathbf{R}^3$ (you may assume it to be uniformly continuous, if you like). Do we have that $$\lim_{\epsilon\rightarrow ...
1
vote
1answer
169 views

error of the composite trapezoidal rule

Let $f\in C^2[a,b]$. The composite trapezoidal rule is given by $$T_n[f]:=h\left(\frac{f(a)+f(b)}{2}+\sum_{k=1}^{n-1}f(x_k)\right)\;\;\;\;\;\left(h:=\frac{b-a}{n},\;x_k:=a+kh\right)$$ First, I've ...
0
votes
2answers
80 views

How do we prove the error estimation of the rectangle method

Let $f\in C^2[a,b]$. An approximation of the integral over $[a,b]$ is given by $$I[f]:=\int_a^bf(x)\text{ dx}\approx \frac{b-a}{n}\sum_{i=1}^nf\left(a+\frac{2i-1}{n}(b-a)\right)=:M_n[f]$$ I've spent ...
4
votes
1answer
197 views

Estimating the sum $\sum_{k=2}^{\infty} \frac{1}{k \ln^2(k)}$

By integral test, it is easy to see that $$\sum_{k=2}^{\infty} \frac{1}{k \ln^2(k)}$$ converges. [Here $\ln(x)$ denotes the natural logarithm, and $\ln^2(x)$ stands for $(\ln(x))^2$] I am ...
2
votes
1answer
60 views

Some estimate concerning hyperbolic functions

I want to show that $|\sinh(az)|\leq|\sinh(z)|$ for all $z\in\mathbf{C}$ (or at least for all $z\in\mathbf{H}$, the upper half plane), provided that $0<a<1$ However, I am not even certain ...
2
votes
1answer
107 views

Prove $ 1 \lt \int_0^1 \frac{1+x^{30}}{1+x^{60}} \ \mathrm{d}x \lt 1 + \frac{1}{30}$ [duplicate]

Possible Duplicate: How to prove $\int_0^1 \frac{1+x^{30}}{1+x^{60}} dx = 1 + \frac{c}{31}$, where $0 \lt c \lt 1$ How can I prove the estimate $$ 1 \lt \int_0^1 \frac{1+x^{30}}{1+x^{60}} \ ...
2
votes
2answers
76 views

Error Term in Passing from Summation to Integral

I encountered the following in a paper and do not understand how the error term is being bounded. In what follows, $n$ and $k$ are large integer constants. $$ \sum_{i=0}^{n-1} \ln\left(1 - ...