Higher Level Integration Marathon & Questions (1 Viewer)

seanieg89

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Re: Extracurricular Integration Marathon

I did think of an idea that probably deals with it though, and will post it during an afternoon break today if it works.
Have been a bit busy the last couple of days, hence the delay. The method does indeed work and is based on the estimate:



if p monotonically decreases to zero and the integral of phi between mT and (m+1)T is const.(-1)^m.

(This comes from the alternating nature of the integrand and is a good exercise to prove.)

Equipped with this result we can truncate our integrals upper limit to m*pi with an explicit error term of O(1/m).

Now that we are working with finite integral, we Taylor expand e^(e^(ix))-e^(e^(-ix)) in powers of e^(ix) to m-th order. As we are evaluating the functions e^z only on the unit disk, we incur an O(1/(m+1)!) error here.

Of course, the interval of integration has length of order m, so overall this thing gives us an O(m/(m+1)!) error. The remaining main term is



The integral occuring in the sum is pi/2, up to an order (1/(k*m*pi)) order error. (We have evaluated the integral earlier in this thread, and the error bound comes because it is of the oscillatory type discussed in this post.)

Summing 1/(k.k!) over ALL of the integers is convergent, so this error term sums to O(1/m) as well. The final error term comes from truncating the Taylor sum for e. This error obviously tends to zero as well, so we are left with the main term, which is simply (e*pi)/2.
 

seanieg89

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Re: Extracurricular Integration Marathon

Note: The above idea is similar in spirit to how we can bound sin/cos between their partial Taylor series of opposite parity.
 

seanieg89

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Re: Extracurricular Integration Marathon

Without invoking the Dilogarithm, or using infinite sums of power series, evaluate

I know you asked to avoid the use of power series, so I will figure out another method for the simpler integral where I use it.

In the meantime though:

 
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Paradoxica

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Re: Extracurricular Integration Marathon

I know you asked to avoid the use of power series, so I will figure out another method for the simpler integral where I use it.

In the meantime though:

Normally I would just accept that you skip from the integral straight to the zeta value but whatever floats your boat :)
 

seanieg89

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Re: Extracurricular Integration Marathon

Just to demonstrate that it is indeed an alternating version of the Basel sum, to people who would not recognise it.

(I certainly don't know/remember that many of these identities off the top of my head myself.)
 

leehuan

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Re: Extracurricular Integration Marathon

Lol.

Double integrals. One of the few things 1251 students have over 1241.

In this regard doing actuarial was not a mistake.
 

Paradoxica

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Re: Extracurricular Integration Marathon

How do I convert this integral directly into the Basel Problem?

Alternatively, Dilogarithms are acceptable.

Contouring would probably work but try to avoid that for now.

 

Paradoxica

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Re: Extracurricular Integration Marathon

How do I convert this integral directly into the Basel Problem?

Alternatively, Dilogarithms are acceptable.

Contouring would probably work but try to avoid that for now.

This may or may not be of use, but :

 
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