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# My meager introduction to the field of mathematics

Discussion in 'Physical & Life Sciences' started by Elendur, Aug 29, 2014.

1. ### ElendurGamer and mathematician

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Hello. My name is André Berglund and I present to you my bachelors essay. This has stemmed from my studies at Umeå University, more specifically the program "Physics and Applied Mathematics".

On to the title (which here will be wonderfully formatted using skills beyond everyone else but me *cough*):
$image=http://latex.codecogs.com/gif.latex?Evaluating&space;\int_0^\infty&space;f(x)dx~and&space;\int_a^b&space;f(x)dx~using~residue~calculus.&hash=ab86634c55a3c267dcf083927b6edffb$

I.e. I'm taking a detour through complex analysis to calculate the real integrals (of arbitrary distance). I will try to answer questions to the best of my abilities. The essay is written such that students who've had an introductory course in complex analysis will be able to, with a bit of extra work, understand the report reading from start to finish.

This thread will also serve as a line in my signature, as I am immensely proud over this essay. I've never done anything as tangible and comprehensive as this before so one might say (crassly) that this is what I've amounted to until now.

PDF available at:
DiVA (official place to go)
(Would have loved to attach it to my OP, though it's essentially only images that are allowed)

I'm also willing to provide additional mirrors, should there be anyone who wants another source (I'd then prefer a suggestion of site).

Last edited: Oct 7, 2014

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Congratulations!

3. ### ElendurGamer and mathematician

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Thank you very much

4. ### MichaelContributorSupporter

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40+ pages! Holy smokes, that's going to take forever to read!

Congrats!

5. ### ElendurGamer and mathematician

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Believe me when I claim:
Not as long as it took for me to write... It has changed so many times.

But thanks

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7. ### godenver1Guest

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What career are you looking toward, if any, Elendur?

8. ### ElendurGamer and mathematician

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I'm at the moment working towards a masters (two additional years), though I don't know whether it will be within statistics or numerical methods at the moment. I will tailor the masters towards a post-graduate doctors position (if that's the correct translation) and after that try to fill a position at the university as a lector (25% research, 75% education).
If that fails, I will work towards teaching mathematics at what we call a gymnasium, ages ~16-19.

So, in short, deeper into mathematics and then teaching . Thanks for asking

9. ### godenver1Guest

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I'm sure if you pass everything you're going for you'll be very qualified! You must have quite a passion for maths, which Is probably a great quality as a teacher.

10. ### ElendurGamer and mathematician

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That's what I'm hoping for as well thank you!
I don't know whether I have a passion for math, but I do have a passion for details and generality which goes together with math perfectly. This also leads me to an interest in programming, logic and philosophy (all of which I've dabbled in but they're not as fun).

Then again, now that I think about it, when I discuss math with others I might well fall into the definition of passionate

11. ### MikeCarraGuest

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Congrats! And at the bachelors level! I assume you are heading off to grad school? Great things await you!

(Mike says with envy for this level of mathematical skill)

12. ### bhsmteNewbie

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Lot's of hard work, congrats!

13. ### ElendurGamer and mathematician

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Indeed, although it's at the same place. I've already started my advanced studies but I'll say this, I don't even compare to some people at school... Intimidating...

It was and thank you

14. ### mzunguINVICTUS

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Bravo! My sincere congratulations

15. ### ElendurGamer and mathematician

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And my sincere thanks

16. ### MichaelContributorSupporter

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I have a question for your Elendur. The small portion of your paper I've had time to read is great. I have a daughter that is taking a first semester Calculus class, and I'm thinking of sending your paper to her. My 'fear" is that she's just studying limits of equations at the moment and I don't want to overwhelm her with a 40+ whopper of paper. It might just scare her and do more harm than good at the moment. I'll wait awhile to send it and be sure she's far enough along to follow along. I'm wondering when you might suggest that I introduce her to your work in terms of how far along in the semester/year? Are their any "signs" along the way that you might suggest before I overwhelm her all at once? Then again, maybe I should be thinking in terms of *years* (plural) before springing something like that on her?

17. ### ElendurGamer and mathematician

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That depends on what calculus course she is reading. We at our university have three essential courses of calculus:
One variable 1 (most likely the one she's reading, given your description)
One variable 2
Multi variable

She should definitely have read a bit of multi variable calculus before this. Just to get a feel of how to think with two variables. This should add up to 3/4 semester (in our speed, it's possible it's faster or slower).

She doesn't have to go for a deep dive in the multi variate (there's usually a whole lot which is irrelevant to this specific topic). But there is one additional step afterwards, getting introduced to complex analysis. To see how different it is to have an imaginary variable as the second dimension instead of a real one (and yes, those are the terms).

I've written this paper such that every step is explicitly stated. Therefore, if the reader is able to pick up at the start, the whole paper will follow, given enough time. Referencing "Calculus, written by Adams, 7th ed" The start would include:
Limits (ch 1.2-1.4)
Derivatives and integrals (ch 2.2-2.4)
Integration (ch 5.3-5.6)
Improper integrals (ch 6.5)
Parametric curves (ch 8.2)
Arc lengths (ch 7.3 , 8.4)
Polar coordinates. (ch 8.5)
Series representation (ch 9.1-9.6)

With the possible exception of parametric curves and arc lengths, they should all be covered by the single variable courses. The exceptions should be covered by multi variable courses. Concepts which are not necessarily included in those courses are:
Open sets.
Complex numbers.
Complex logarithm. (Though I mostly skip this one it can help with the thinking of the last theorems)

There are also two skills which take time to develop, and it's very individual when one reaches the point when one's sufficiently proficient:
The thinking of definitions.
The thinking of curves.

The most important part is the skills. Hands down. The skills to understand the curve choices and to understand how to use definitions.

If you have more questions, I'm more than happy to elaborate further.

18. ### MichaelContributorSupporter

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Thanks. Awesome work.

19. ### katerinah1947Well-Known MemberSupporter

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Hi,
I am entirely upset with you. And no, I don't want you to bother me, with any of your undergraduate excuses. Please explain to me: WHY you were not my math teacher. Yes, I went college in 1966, and as I said earlier, no excuses. You could have been born earlier, but YOU CHOSE not to, didn't you?
Okay, I'll forgive you this time.
To infinity?. Nice,.

20. ### ElendurGamer and mathematician

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I was told you would do fine without me

To infinity. And beyond! (Not really beyond, but it's catchy )