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#26 Help Me ! » complex function » 2009-04-26 00:03:51

coffeeking
Replies: 4

Hi, please check if I have made any mistake.

Determine if

contain a primitive on
.

Below is my working:


has primitive
where

Since

has analytic domain

where

Therefore,

does not has a primitive on
.

Thanks in advance.

#27 Re: Help Me ! » Indented Contour » 2009-04-19 16:18:07

coffeeking

The contour

for this question goes something like:


                            __<__
                           /         \
                          | 2(pi)i  |
-R+2pi(i) -----<---    .       -----<-------  R+2pi(i)
              |                                        |
              |                                        |
              |                                        |
              v                                       ^
              |                                        |
              |                                        |
        -R   |______>   .     ____>_____| R
                          |    0    |
                           \__>_/


Where the contour around

and and
is a semi-circular arc.

#28 Help Me ! » Indented Contour » 2009-04-19 05:41:11

coffeeking
Replies: 3

Hi, below is a question on indented contour where I have the solution but don't understand what's going on.

The question goes like this :

Find


and the solution goes something like:


By defining indented contour on a rectangle contour
around the point
and


Well my question is why we take only half the residue on

and
? Isn't it by Cauchy's Residue Theorem that for a closed positively oriented contour

#29 Re: Help Me ! » Chain rule » 2009-04-19 04:22:00

coffeeking





#30 Re: Help Me ! » Complex Integration » 2009-03-16 15:16:16

coffeeking

hmm.. Think that should clear things up thanks Ricky. However, there is another thing about Laurent Series that I don't understand

Laurent Series states that:

Let

be analytic in the annulus
. Then
can be expressed there as

Where

is given by:

Well, same confusion here... The first statement with the summation sign are all in

and
however, the second statement that tell us about
are in
... So what
actually means in this context?

#31 Help Me ! » Complex Integration » 2009-03-16 05:26:54

coffeeking
Replies: 3

Well, I have the solution to the problem below, just that I don't understand what is going on dunno

Let

be a simple closed positively oriented contour that passes through the point
. Set

Find

, where z approaches
from inside

The solution given is:

Since

is analytic on
, by Cauchy's Integral Formula,

for
inside

Therefore

by continuity of cosine function.

Well, the thing I don't understand is why

? I don't see any link of it to Cauchy Integral Formula, also what
is refering to in this context?

In addition I am totally confuse of why we replace

by
and
by
in Cauchy Integral Formula as I don't understand what is the rationale of doing so and more importantly what we mean by
and integration with respect to
in this context.

I will be more than grateful if any kind soul could explain this to me as I have been struggling to understand many of the concepts even since I taken this course...

PS. Sorry if my question sound stupid or ridiculously easy... I am just someone who are trying his very best to learn...

Thanks in advance. smile

#32 Re: Help Me ! » Regarding Cubic Outline Problem under Measuring » 2009-03-10 00:53:37

coffeeking

Yes if you are talking about combination (i.e the order doesn't matter).

#33 Re: Help Me ! » How do I differentiate this function? » 2009-03-05 05:25:58

coffeeking

Well, if you wish




#34 Re: Help Me ! » How do you do this integration? » 2009-03-03 04:34:07

coffeeking



#35 Re: Help Me ! » Uniform Continuity » 2009-03-02 06:34:55

coffeeking

Hi Jane, thanks for helping, I appreciate it. Well, I do agree with you that I should feel ashamed of myself for asking such a simple question, in fact I already did neutral...

Actually even since my school term started I have met a lot of difficulties trying to understand this course as my lecture notes are really wordy, lack of illustration and worked examples and full of typo errors. Furthermore as to make the matter worse, my lecturer is not being clear in his explanation. In fact, a lot of his students including me feel even more doubtful after asking him question in the hope that he could clear our doubts.

Currently I have borrowed lecture notes from my friends from other university, which seem to be 100 times better than mine and I have been trying my very best to catch up. I became more and more frustrated after looking through my friends' notes and realised my lecturer has left out a great deal of definitions and illustrations in his notes. An example is that he did not even explain to us what is a simply connected domain when Cauchy's Integral Theorem was being taught.

Frankly speaking, when I was posting this question I am actually finding "the easy way out" as I have too much stuffs to catch up lately. I was hoping that I could in fact learn from the working posted in the forum as I find it easier to learn through worked example where my lecture notes really lacks.

I can say that I have learned something new today and I have to really thanks you again for your willingness to help smile and don't worry, I understand your working.:P

Well, as for the second question I think I have figured out how to do. Below is my solution please check my working for me:

Suppose 

for

then       

             

             

             

Therefore choose

Hence

is uniformly continuous.

In addition I have another question that I am unsure of. My question is is it true that epsilon delta method when use to prove continuity we can only prove continuity at a point and not the whole function unlike for uniform continuity?

Thanks in advance. smile

#36 Help Me ! » Uniform Continuity » 2009-03-01 18:55:56

coffeeking
Replies: 2

Which of these functions (or both) is/are uniformly continuous and why?


Sorry if my question sound noobish... Thanks in advance.:D

#37 Re: Help Me ! » determine the slope of the tangent when x coordinate is a » 2009-02-19 14:46:19

coffeeking

Ok, starting from


Therefore

#38 Re: Help Me ! » Prove this » 2009-02-17 20:10:00

coffeeking

Oh my gosheek... Think I need more coffee tongue you are right meo_beo, I have made a mistake sorry for the confusion.

Anyway, Dragonshade has a great proof. However if I observed correctly this time round, I think the last line of your proof should be

Since from

there are a total of
terms.

#39 Re: Help Me ! » Prove this » 2009-02-17 16:00:43

coffeeking

Well, I think you get the inequality sign for the question wrong.

Assuming what you type is

Observe that

Therefore,

#40 Re: Help Me ! » limits » 2009-02-15 02:36:25

coffeeking

Bump this up since I am interested in knowing too...

#41 Re: Help Me ! » more basic limits - cant get the answer that it states » 2009-02-07 00:25:59

coffeeking

Well, this is how I solve the problem





#42 Re: Help Me ! » Help in mathematical symbol... » 2009-02-06 23:52:38

coffeeking

Hi Jane, I think you have hit the nail on the head and I have finally understood what it means.up

Thanks smile

#43 Help Me ! » Help in mathematical symbol... » 2009-02-06 21:08:17

coffeeking
Replies: 2

Hi, I am an undergraduate currently taking a course in Complex Analysis. As my foundation in Sets Theory is not strong, I have problem understanding some of the expression used.dunno

Below is a statement about countable compactness in Complex Analysis:

"A family F of subsets of C is said to cover S if

...

My question is, what does the expression

means?

From what I have learn, I read it as "S is a subset of the union of the set U which is an element of F" which sound confusing as we always use "

" to denote union of two things but now it's used to denote an union of something which is an element of something. dizzy

I will appreciate if anyone could explain it to me in the most layman term where possible.

Thanks in advance.

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