Counter Integration! (1 Viewer)

[U MAD BRO]

Carrot, can you teach me how to do contour integration, looks like a pretty cool technique.
For example, I'm doing this bad ass nasty little integral and I couldn't continue because I have a contour integral:








then consider the contour integral:





where the contour is the half-disc in the upper half plane, then just use the Residue Theorem.


I think this is really interesting, I never thought integration gets any more advanced =)

 

[zeebobDD]

yeah i'll teach you man, you just multiply the top and bottom by (z-i)(z+i) then a few little manipulations and you'll arrive at your answer

 

[SpiralFlex]

Counter integration? You mean contour?

 

[Carrotsticks]

1. It's 'Contour Integration'.

2. You need basic knowledge of Topology and open/closed sets and/or balls in some n'th dimension (although 'balls' are specifically 3D, we use it as a general term for the n'th dimensional equivalent of a sphere)

3. Basic knowledge of Real Analysis is CRUCIAL, despite us working in the Complex Field because the theorems and tests (ie: Weierstrass M-Test) are very much applicable to the proofs of theorems used in Complex Analysis, such as the Residue Theorem.

4. Although Laurent expansions at the basic level can be done by the average high-school student, the proofs required to justify their convergence for use in Cauchy's Theorem are by no means accessible at an elementary level.

5. Yes, Residues can be applied to real indefinite integrals but you need a fairly solid knowledge of contour paths and analytic functions + singularities/poles before moving on to using them properly.

So essentially, you have about a semester or two of things to learn before you can properly move on to tackling these kinds of problems.

 

[U MAD BRO]

1. It's 'Contour Integration'.

2. You need basic knowledge of Topography and open/closed sets and/or balls in some n'th dimension (although 'balls' are specifically 3D, we use it as a general term for the n'th dimensional equivalent of a sphere)

3. Basic knowledge of Real Analysis is CRUCIAL, despite us working in the Complex Field because the theorems and tests (ie: Weierstrass M-Test) are very much applicable to the proofs of theorems used in Complex Analysis, such as the Residue Theorem.

4. Although Laurent expansions at the basic level can be done by the average high-school student, the proofs required to justify their convergence for use in Cauchy's Theorem are by no means accessible at an elementary level.

5. Yes, Residues can be applied to real indefinite integrals but you need a fairly solid knowledge of contour paths and analytic functions + singularities/poles before moving on to using them properly.

So essentially, you have about a semester or two of things to learn before you can properly move on to tackling these kinds of problems.

Wow, didn't think it is this complicated.

 

[math man]

Now i wrote up the solution, i doubt you will understand it all, but here is the method used to prove it:

 

[seanieg89]

1. It's 'Contour Integration'.

2. You need basic knowledge of Topography and open/closed sets and/or balls in some n'th dimension (although 'balls' are specifically 3D, we use it as a general term for the n'th dimensional equivalent of a sphere)

3. Basic knowledge of Real Analysis is CRUCIAL, despite us working in the Complex Field because the theorems and tests (ie: Weierstrass M-Test) are very much applicable to the proofs of theorems used in Complex Analysis, such as the Residue Theorem.

4. Although Laurent expansions at the basic level can be done by the average high-school student, the proofs required to justify their convergence for use in Cauchy's Theorem are by no means accessible at an elementary level.

5. Yes, Residues can be applied to real indefinite integrals but you need a fairly solid knowledge of contour paths and analytic functions + singularities/poles before moving on to using them properly.

So essentially, you have about a semester or two of things to learn before you can properly move on to tackling these kinds of problems.

Topography hey? .

 

[Carrotsticks]

Haha never noticed. Fixed now =)

Kept thinking 'contours' so naturally 'topography' came up in my head instead of topology.

 

[math man]

lmao tography, our very own geographicist

 

[Carrotsticks]

How's your lim inf Analysis score? =)

 

[U MAD BRO]

Now i wrote up the solution, i doubt you will understand it all, but here is the method used to prove it:

View attachment 26096
View attachment 26097

I do understand it =) I've been reading about the theory behind it for the past 2 hours and yes, the answer is pi/2e
thanks for the solution maths man will post more questions in extra curricular from now on.

 

[math man]

How's your lim inf Analysis score? =)

looking as good as linear

 

[U MAD BRO]

Carrotsticks, visit this website and tell me if you like the idea =]
http://mathim.com/Mathstuff
a LaTeX chat, we should have one here so we can learn maths faster !!!

 

[U MAD BRO]

also can u fix the title for me