Autoparallel: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
en>JHSnl
Added {{unreferenced}} tag to article (TW)
 
en>Slawekb
It's only a stub. Stubs are by their nature incomplete.
Line 1: Line 1:
The art of home improvement is so much more than just putting in a new sink. It takes a great deal of research, a lot of hard work, and careful attention to detail to really make sure that things look nice and that they are safely installed. The tips below can help you improve your home improvement skills.<br><br>Spend some time in your prospective neighborhood driving and walking through. You should have some idea about the kind of maintenance your new neighbors apply to their homes. If the area seems run down with overgrown yards and untidy homes, you will want to rethink moving into the area.<br><br>To avoid costly mistakes when painting a room, make a small investment in the paint samples that many retailers offer. You can get 8 ounce sample sizes in any available color. For a small cost, you can apply paint to a big enough area to really get a feel for how the paint will look, much more accurately than with paint chips alone.<br><br>Drive nail holes in the rim of your paint can! The channel near the top can fill, when replacing the lid that paint is pushed up and over the paint can's sides. Using a nail you can add holes around the can's perimeter of the channel and this will fix the issue.<br><br>It is a very good idea to own your home. Most people want to decorate their house and make improvements to fit their lifestyle, but when you rent you need to ask for permission to make certain improvements. It is much better to do that in your own property, as it doesn't make any sense to spend thousands of dollars to improve someone else's property.<br><br>A great way to conserve energy while also keeping bugs out of your home, is to seal up any cracks in your home. To fix those window gaps, you can get some simple caulk. For gaps or holes in the wall or floorboard, you can try Spackle or canned foam. After the caulk has dried, bugs should have a hard time getting in, and air should have a hard time escaping.<br><br>When you hire a contractor for your project, it's necessary to keep a file of all your records. Do not rely on the contractor to handle this for you. Save all the paperwork that is at all related to your project. This can help you and the contractor stay focused on the current tasks.<br><br>If you have an extensive collection of collectibles or figurines, keep what you display to a minimum. Trying to display everything in one room or area can make your space appear cluttered. Choose a few focal pieces, about 5 pieces at the most and arrange them in a pleasing manner.<br><br>To nicely display your jewelry, set up a straight coat rack that can be mounted to a wall. Only put costume jewelry here, not your prized heirlooms. Hanging your jewelry on a wall can make a pretty decorative statement, as well as keeping the jewelry tangle-free. Choose several items that you wear a lot and display those.<br><br>Check your pipes and plumbing regularly. Clogs in sinks, tubs, and toilets can lead to [http://Search.Usa.gov/search?query=water+back-ups water back-ups]. Flooding from back-ups can create an awful mess and cause massive amounts of expensive damage. Slow moving drains can cause rings of dirt to appear in your tubs and sinks, creating a embarrassing issue. It is best to keep on top of and remove clogs regularly.<br><br>Clean kitchen counters can help to sell a house. Remove everything except your coffee pot, a nice set of knives, and perhaps a bowl of fruit. The cleaner your counter tops are, the more organized your kitchen will appear. Your kitchen will also feel larger and more user friendly.<br><br>Many people think a kitchen remodel can be time consuming and expensive. However, there are many things a [http://Www.Bbc.Co.uk/search/?q=homeowner homeowner] on a limited budget can do to change the look of their kitchen. By simply replacing countertops, updating appliances or refinishing cabinets, a homeowner can give their kitchen a facelift without breaking the bank.<br><br>The art of home improvement is something that can be enjoyed by nearly everyone, but only those very serious will try to perfect their home like a true professional. Now with more home improvement knowledge to add to your "bag of tricks," you can easily become a great home improver too.<br><br>If you adored this article and you would such as to get additional information relating to [https://www.rebelmouse.com/dizzytailor4957/great-roofing-tips-you-should--772173738.html home automation] kindly go to our site.
In [[mathematics]], '''Ramanujan's master theorem''' (named after mathematician [[Srinivasa Ramanujan]]<ref>B. Berndt. Ramanujan’s Notebooks, Part I. Springer-Verlag, New York, 1985.</ref>) is a technique which provides an analytic expression for the [[Mellin transform]] of a function.
[[Image:Ramanujanmasterth.jpg|thumb|Page from Ramanujan's notebook stating his Master theorem.]]
 
The result is stated as follows:
 
Assume function <math> f(x) \!</math> has an expansion of the form
 
: <math> f(x)=\sum_{k=0}^\infty \frac{\phi(k)}{k!}(-x)^k \!</math>
 
then [[Mellin transform]] of <math> f(x) \!</math> is given by
 
: <math> \int_0^\infty x^{s-1} f(x) \, dx = \Gamma(s)\phi(-s) \!</math>
 
where <math> \Gamma(s) \!</math> is the [[Gamma function]].
 
It was widely used by Ramanujan to calculate definite integrals and [[infinite series]].
 
Multidimensional version of this theorem also appear in [[quantum physics]] (through [[Feynman diagram]]s).<ref>[http://129.81.170.14/~vhm/papers_html/RMT-GMS.pdf A generalized Ramanujan Master Theorem applied to the evaluation of Feynman diagrams by Iv´an Gonz´alez, V. H. Moll and Iv´an Schmidt]</ref>
 
A similar result was also obtained by [[J. W. L. Glaisher]].<ref>J. W. L. Glaisher. A new formula in definite integrals. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 48(315):53–55, Jul 1874.</ref>
 
== Alternative formalism ==
An alternative formulation of Ramanujan's master theorem is as follows:
 
: <math> \int_0^\infty  x^{s-1} ({\lambda(0)-x\lambda(1)+x^{2}\lambda(2)-\cdots}) \, dx = \frac{\pi}{\sin(\pi s)}\lambda(-s) </math>
 
which gets converted to original form after substituting <math> \lambda(n) = \frac{\phi(n)}{\Gamma(1+n)} \!</math> and using functional equation for [[Gamma function]].
 
The integral above is convergent for <math> 0< \operatorname{Re}(s)<1 \!</math>.
 
== Proof ==
The proof of Ramanujan's Master Theorem provided by [[G. H. Hardy]]<ref>G. H. Hardy. Ramanujan. Twelve Lectures on subjects suggested by his life and work. [[Chelsea Publishing Company]], New York, N. Y., 3rd edition, 1978.</ref> employs [[Cauchy residue theorem|Cauchy's residue theorem]] as well as the well-known [[Mellin inversion theorem]].
 
== Application to Bernoulli polynomials ==
The generating function of the [[Bernoulli polynomials]] <math> B_k(x)\!</math> is given by:
 
: <math> \frac{ze^{xz}}{e^z-1}=\sum_{k=0}^\infty  B_k(x)\frac{z^k}{k!} \!</math>
 
These polynomials are given in terms of [[Hurwitz zeta function]]:
: <math> \zeta(s,a)=\sum_{n=0}^\infty  \frac{1}{(n+a)^s} \!</math>
 
by <math> \zeta(1-n,a)=-\frac{B_n(a)}{n} \!</math> for <math> n\geq1 \!</math>.
By means of Ramanujan master theorem and generating function of Bernoulli polynomials one will have following integral representation:<ref>O. Espinosa and V. Moll. On some definite integrals involving the Hurwitz zeta function. Part 2. The Ramanujan Journal, 6:449–468, 2002.</ref>
 
: <math> \int_0^\infty  x^{s-1} \left(\frac{e^{-ax}}{1-e^{-x}}-\frac{1}{x}\right) \, dx = \Gamma(s)\zeta(s,a) \!</math>
 
valid for <math> 0<Re(s)<1\!</math>.
 
== Application to the Gamma function ==
Weierstrass's definition of the Gamma function
 
: <math> \Gamma(x)=\frac{e^{-\gamma x}}{x}\prod_{n=1}^\infty \left(1+\frac{x}{n}\right)^{-1} e^{x/n} \!</math>
 
is equivalent to expression
 
: <math> \log\Gamma(1+x)=-\gamma x+\sum_{k=2}^\infty \frac{\zeta(k)}{k}(-x)^k \!</math>
 
where <math> \zeta(k) \!</math> is the [[Riemann zeta function]].
 
Then applying Ramanujan master theorem we have:
 
: <math> \int_0^\infty x^{s-1} \frac{\gamma x+\log\Gamma(1+x)}{x^2} \, dx= \frac{\pi}{\sin(\pi s)}\frac{\zeta(2-s)}{2-s} \!</math>
 
valid for <math> 0<Re(s)<1\!</math>.
 
Special cases of <math> s=\frac{1}{2} \!</math> and <math> s=\frac{3}{4} \!</math> are
 
: <math> \int_0^\infty \frac{\gamma x+\log\Gamma(1+x)}{x^{5/2}} \, dx =\frac{2\pi}{3} \zeta\left( \frac{3}{2} \right) </math>
 
: <math> \int_0^\infty \frac{\gamma x+\log\Gamma(1+x)}{x^{5/4}} \,dx = \sqrt{2} \frac{4\pi}{5} \zeta\left(\frac 5 4\right) </math>
 
[[Mathematica|Mathematica 7]] is unable to compute these examples.<ref>Ramanujan's Master Theorem by Tewodros Amdeberhan, Ivan Gonzalez, Marshall Harrison, Victor H. Moll and Armin Straub, The Ramanujan Journal.</ref>
 
== Evaluation of quartic integral ==
It is well known for the evaluation of
 
: <math> F(a,m)=\int_0^\infty \frac{dx}{(x^4+2ax^2+1)^{m+1}} </math>
which is a well known quartic integral.<ref>T. Amdeberhan and V. Moll. A formula for a quartic integral: a survey of old proofs and some new ones. The Ramanujan Journal, 18:91–102, 2009.</ref>
 
== References ==
{{Reflist}}
 
== External links ==
# http://mathworld.wolfram.com/RamanujansMasterTheorem.html
# http://www.youtube.com/watch?v=gLp5OsfUlNE
# http://arminstraub.com/files/publications/rmt.pdf
 
[[Category:Srinivasa Ramanujan]]
[[Category:Theorems in analytic number theory]]

Revision as of 13:05, 16 January 2013

In mathematics, Ramanujan's master theorem (named after mathematician Srinivasa Ramanujan[1]) is a technique which provides an analytic expression for the Mellin transform of a function.

File:Ramanujanmasterth.jpg
Page from Ramanujan's notebook stating his Master theorem.

The result is stated as follows:

Assume function f(x) has an expansion of the form

f(x)=k=0ϕ(k)k!(x)k

then Mellin transform of f(x) is given by

0xs1f(x)dx=Γ(s)ϕ(s)

where Γ(s) is the Gamma function.

It was widely used by Ramanujan to calculate definite integrals and infinite series.

Multidimensional version of this theorem also appear in quantum physics (through Feynman diagrams).[2]

A similar result was also obtained by J. W. L. Glaisher.[3]

Alternative formalism

An alternative formulation of Ramanujan's master theorem is as follows:

0xs1(λ(0)xλ(1)+x2λ(2))dx=πsin(πs)λ(s)

which gets converted to original form after substituting λ(n)=ϕ(n)Γ(1+n) and using functional equation for Gamma function.

The integral above is convergent for 0<(s)<1.

Proof

The proof of Ramanujan's Master Theorem provided by G. H. Hardy[4] employs Cauchy's residue theorem as well as the well-known Mellin inversion theorem.

Application to Bernoulli polynomials

The generating function of the Bernoulli polynomials Bk(x) is given by:

zexzez1=k=0Bk(x)zkk!

These polynomials are given in terms of Hurwitz zeta function:

ζ(s,a)=n=01(n+a)s

by ζ(1n,a)=Bn(a)n for n1. By means of Ramanujan master theorem and generating function of Bernoulli polynomials one will have following integral representation:[5]

0xs1(eax1ex1x)dx=Γ(s)ζ(s,a)

valid for 0<Re(s)<1.

Application to the Gamma function

Weierstrass's definition of the Gamma function

Γ(x)=eγxxn=1(1+xn)1ex/n

is equivalent to expression

logΓ(1+x)=γx+k=2ζ(k)k(x)k

where ζ(k) is the Riemann zeta function.

Then applying Ramanujan master theorem we have:

0xs1γx+logΓ(1+x)x2dx=πsin(πs)ζ(2s)2s

valid for 0<Re(s)<1.

Special cases of s=12 and s=34 are

0γx+logΓ(1+x)x5/2dx=2π3ζ(32)
0γx+logΓ(1+x)x5/4dx=24π5ζ(54)

Mathematica 7 is unable to compute these examples.[6]

Evaluation of quartic integral

It is well known for the evaluation of

F(a,m)=0dx(x4+2ax2+1)m+1

which is a well known quartic integral.[7]

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

  1. http://mathworld.wolfram.com/RamanujansMasterTheorem.html
  2. http://www.youtube.com/watch?v=gLp5OsfUlNE
  3. http://arminstraub.com/files/publications/rmt.pdf
  1. B. Berndt. Ramanujan’s Notebooks, Part I. Springer-Verlag, New York, 1985.
  2. A generalized Ramanujan Master Theorem applied to the evaluation of Feynman diagrams by Iv´an Gonz´alez, V. H. Moll and Iv´an Schmidt
  3. J. W. L. Glaisher. A new formula in definite integrals. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 48(315):53–55, Jul 1874.
  4. G. H. Hardy. Ramanujan. Twelve Lectures on subjects suggested by his life and work. Chelsea Publishing Company, New York, N. Y., 3rd edition, 1978.
  5. O. Espinosa and V. Moll. On some definite integrals involving the Hurwitz zeta function. Part 2. The Ramanujan Journal, 6:449–468, 2002.
  6. Ramanujan's Master Theorem by Tewodros Amdeberhan, Ivan Gonzalez, Marshall Harrison, Victor H. Moll and Armin Straub, The Ramanujan Journal.
  7. T. Amdeberhan and V. Moll. A formula for a quartic integral: a survey of old proofs and some new ones. The Ramanujan Journal, 18:91–102, 2009.