quite a late reply
I write this answer on 2013, so it is quite late but I still hope this be helpful.
I know a book that can definitely satisfy you.
It is Advanced Calculus by Fitzpatrick.
I've recently read Munkres's and Duistermaat's.
The former one is very very good. It is...
Thanks mathwonk. I will definitely keep in mind your insight and comment when I study this in a more rigorous setting.
And thanks Bacle2. Indeed the quotient topology section in Munkres seems quite good; it does not use that fluffy method in the proof.
Thanks for the reply mathwonk! I actually thought I wouldn't get any answer, anyway thanks. Okay as for the textbooks that you'v recommneded, I would look them up. But I'm very new to this topic. So obviously I haven't read them at all. Actually I'm not even really studying 'real' algebraic...
I was working on some algebraic topology matters, thinkgs like the connected sum of some surfaces is some other surface. And for this study, I was using the Munkres's famous textbook 'Topology' the algebraic topology part. My qeustions are as follows:
Q1) Munkres introduces 'labelling scheme'...
Thanks for your answer, and as you have told me I should assume that f(x) \geq 0 a.e.. And for the last part you mentioned, I think Dominated Convergence Theorem would suffice, woudn't it? I used it in my solution..
Homework Statement
Let ( \mathbb{R}^k , \mathcal{A} , m_{k} ) be a Lebesgue measurable space, i.e., m_{k}=m is a Lebesgue measure. Let f: \mathbb{R^k} \to \mathbb{R} be a m-integrable function. Define a function \mu : \mathcal{A} \to [0,\infty] by $$ \mu(A) := \int_{A} f(x) dx $$ with A \in...
Ah.. frankly I'm not sure of how to expand by Taylor series... Is it by using Cauchy's Theorem? Could you give me any reference textbook where I can look up the theorem for this Taylor series expansion?
Homework Statement
The wiki page says that error function \mbox{erf}(z) = \int_{0}^{z} e^{-t^{2}} dt is entire. But I cannot find anywhere its proof. Could you give me some stcratch proof of this?
Homework Equations
The Attempt at a Solution
I've tried to use Fundamental Theorem...
Ah... MY BAD! sorry.. what was I thinking... Let me clarify once more:
Take x_{n} \in (-n,1-n) . Then \Gamma (x_{n}) \to 0 as n \to \infty .
I think I have an idea to solve it without using Gauss's Formula. After I try, I will put on the thread.
Anyway thanks for reminding me.
Ah.. I know what you mean. Maybe I need to modify my problem first. I know it has poles on non-positive integers. But excluding poles, it seems the absolute value of the gamma function tends to zero as x \to - \infty .
(http://en.wikipedia.org/wiki/File:Complex_gamma_function_abs.png)
May I...
Homework Statement
The absolute value of the gamma function \Gamma (x) that is defined on the negative real axis tends to zero as x \to - \infty . Right? But how do I prove it?
Homework Equations
The Attempt at a Solution
I've tried to use Gauss's Formula...
Okay..
I think by your notion it seems very plausible to conclude that
\lim_{n\to\infty}\frac{f(z_{0}+h_{n})-f(z_{0})}{h_{n}}=\lim_{h\to0}\frac{f(z_{0}+h)-f(z_{0})}{h}
though I'm not using Thm 4.2 in Baby Rudin
But what about...
Ah of course yes, baby Rudin does not use sequences in his proof (in his latest edition).
And Rudin's proof is very clear whereas Conway's seems not valid.
As for the lemma, doesn't that lemma state that
'limit exists iff for an "arbitrary" sequence the sequential limit exists'...
I ask this question only to those who read or have this book:
If you have Baby Rudin, it would be even better.
On the page 34 of the text Conway's Functions of One Complex Variable Vol 1, it proves the Chain Rule
but it seems the proof is not valid:
It uses sequences to show the limit is...