Two-port network: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older editNewer edit →
en>Catslash
m Interrelation of parameters; make notation for determinant consistent Δz -> Δ[z] and Δ]y] -> Δ[y]
Line 1: Line 1:
In [[mathematics]], '''Wallis' product''' for [[Pi|π]], written down in 1655 by [[John Wallis]], states that
It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br><br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>For more information regarding [http://www.youtube.com/watch?v=90z1mmiwNS8 dentist DC] review our internet site.
 
:<math>
\prod_{n=1}^{\infty} \left(\frac{2n}{2n-1} \cdot \frac{2n}{2n+1}\right) = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \frac{8}{7} \cdot \frac{8}{9} \cdots = \frac{\pi}{2}
</math>
 
==Derivation==
Wallis did not derive this [[infinite product]] as it is done in calculus books today, by comparing <math>\scriptstyle \int_0^\pi \sin^nxdx</math> for even and odd values of ''n'', and noting that for large ''n'', increasing ''n''  by 1 results in a change that becomes ever smaller as ''n'' increases. Since [[infinitesimal calculus]] as we know it did not yet exist then, and the [[mathematical analysis]] of the time was inadequate to discuss the convergence issues, this was a hard piece of research, and tentative as well.
 
Wallis' product is, in retrospect, an easy corollary of the later [[Infinite product#Product representations of functions|Euler formula]] for the [[sine function]].
 
== Proof using Euler's infinite product for the sine function<ref name=WallisFormula>{{cite web|url=http://mathworld.wolfram.com/WallisFormula.html|title=Wallis Formula}}</ref> ==
:<math>\frac{\sin x}{x} = \prod_{n=1}^\infty\left(1 - \frac{x^2}{n^2\pi^2}\right)</math>
 
Let ''x'' = {{frac|π|2}}:
:<math>\begin{align}
  \Rightarrow\frac{2}{\pi} &= \prod_{n=1}^{\infty} \left(1 - \frac{1}{4n^2}\right) \\
  \Rightarrow\frac{\pi}{2} &= \prod_{n=1}^{\infty} \left(\frac{4n^2}{4n^2 - 1}\right) \\
                          &= \prod_{n=1}^{\infty} \left(\frac{2n}{2n-1}\cdot\frac{2n}{2n+1}\right) = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdots
\end{align}
</math>
 
== Proof using integration<ref>{{cite web|url=http://www.sosmath.com/calculus/integration/powerproduct/problem/problem.html|title=Integrating Powers and Product of Sines and Cosines: Challenging Problems}}</ref> ==
 
Let:
:<math>I(n) = \int_0^\pi \sin^nxdx</math>
(a form of the [[Wallis' integrals]]).
Integrate by parts:
:<math>\begin{align}
              u &= \sin^{n-1}x \\
  \Rightarrow du &= (n-1) \sin^{n-2}x \cos x dx \\
              dv &= \sin x dx \\
  \Rightarrow v &= -\cos x
\end{align}</math>
 
:<math>\begin{align}
\Rightarrow I(n) &=  \int_0^\pi \sin^nxdx=\int_0^\pi u dv = uv |_{x=0}^{x=\pi}-\int_0^\pi v du \\
              {} &= -\sin^{n-1}x\cos x |_{x=0}^{x=\pi} - \int_0^\pi - \cos x(n-1) \sin^{n-2}x \cos x dx \\
              {} &= 0 - (n-1) \int_0^\pi -\cos^2x \sin^{n-2}x dx, n > 1 \\
              {} &= (n - 1) \int_0^\pi (1-\sin^2 x) \sin^{n-2}x dx \\
              {} &= (n - 1) \int_0^\pi \sin^{n-2}x dx - (n - 1) \int_0^\pi \sin^{n}x dx \\
              {} &= (n - 1) I(n-2)-(n-1) I(n) \\
              {} &= \frac{n-1}{n} I(n-2) \\
\Rightarrow \frac{I(n)}{I(n-2)}
                  &= \frac{n-1}{n} \\
\Rightarrow \frac{I(2n-1)}{I(2n+1)}
                  &=\frac{2n+1}{2n}
\end{align}</math>
This result will be used below:
 
:<math>\begin{align}
I(0)  &= \int_0^\pi dx = x|_0^\pi = \pi \\
I(1)  &= \int_0^\pi \sin xdx = -\cos x|_0^\pi = (-\cos \pi)-(-\cos 0) = -(-1)-(-1) = 2 \\
I(2n) &= \int_0^\pi \sin^{2n}xdx = \frac{2n-1}{2n}I(2n-2) = \frac{2n-1}{2n} \cdot \frac{2n-3}{2n-2}I(2n-4)
\end{align}</math>
 
Repeating the process,
:<math>=\frac{2n-1}{2n} \cdot \frac{2n-3}{2n-2} \cdot \frac{2n-5}{2n-4} \cdot \cdots \cdot \frac{5}{6} \cdot \frac{3}{4} \cdot \frac{1}{2} I(0)=\pi \prod_{k=1}^n \frac{2k-1}{2k}</math>
 
:<math>I(2n+1)=\int_0^\pi \sin^{2n+1}xdx=\frac{2n}{2n+1}I(2n-1)=\frac{2n}{2n+1} \cdot \frac{2n-2}{2n-1}I(2n-3)</math>
Repeating the process,
:<math>=\frac{2n}{2n+1} \cdot \frac{2n-2}{2n-1} \cdot \frac{2n-4}{2n-3} \cdot \cdots \cdot \frac{6}{7} \cdot \frac{4}{5} \cdot \frac{2}{3} I(1)=2 \prod_{k=1}^n \frac{2k}{2k+1}</math>
 
:<math>\sin^{2n+1}x \le \sin^{2n}x \le \sin^{2n-1}x, 0 \le x \le \pi</math>
:<math>\Rightarrow I(2n+1) \le I(2n) \le I(2n-1)</math>
:<math>\Rightarrow 1 \le \frac{I(2n)}{I(2n+1)} \le \frac{I(2n-1)}{I(2n+1)}=\frac{2n+1}{2n}</math>, from above results.
By the [[squeeze theorem]],
:<math>\Rightarrow \lim_{n\rightarrow\infty} \frac{I(2n)}{I(2n+1)}=1</math>
 
:<math>\lim_{n\rightarrow\infty} \frac{I(2n)}{I(2n+1)}=\frac{\pi}{2} \lim_{n\rightarrow\infty} \prod_{k=1}^n \left(\frac{2k-1}{2k} \cdot \frac{2k+1}{2k}\right)=1</math>
:<math>\Rightarrow \frac{\pi}{2}=\prod_{k=1}^\infty \left(\frac{2k}{2k-1} \cdot \frac{2k}{2k+1}\right)=\frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \cdots</math>
 
== Relation to Stirling's approximation==
 
[[Stirling's approximation]] for ''n''! asserts that
:<math>n! = \sqrt {2\pi n} {\left(\frac{n}{e}\right)}^n \left[1 + O\left(\frac{1}{n}\right) \right]</math>
 
as ''n'' → ∞. Consider now the finite approximations to the Wallis product, obtained by taking the first ''k'' terms in the [[Product (mathematics)|product]]:
:<math>p_k = \prod_{n=1}^{k} \frac{2n}{2n - 1}\frac{2n}{2n + 1}</math>
 
''p<sub>k</sub>'' can be written as
:<math>\begin{align}
  p_k &= {1 \over {2k + 1}} \prod_{n=1}^{k} \frac{(2n)^4}{[(2n)(2n - 1)]^2} \\
      &= {1 \over {2k + 1}} \cdot {{2^{4k}\,(k!)^4} \over {[(2k)!]^2}}
\end{align}</math>
 
Substituting Stirling's approximation in this expression (both for ''k''! and (2''k'')!) one can deduce (after a short calculation) that ''p<sub>k</sub>'' converges to {{frac|π|2}} as ''k'' → ∞.
 
==ζ'(0)<ref name="WallisFormula"/>==
The [[Riemann zeta function]] and the [[Dirichlet eta function]] can be defined:
:<math>\begin{align}
  \zeta(s) &= \sum_{n=1}^\infty \frac{1}{n^s}, \Re(s)>1 \\
  \eta(s)  &= (1-2^{1-s})\zeta(s) \\
          &= \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^s}, \Re(s)>0
\end{align}</math>
 
Applying an Euler transform to the latter series, the following is obtained:
:<math>\begin{align}
 
              \eta(s) &= \frac{1}{2}+\frac{1}{2} \sum_{n=1}^\infty (-1)^{n-1}\left[\frac{1}{n^s}-\frac{1}{(n+1)^s}\right], \Re(s)>-1 \\
  \Rightarrow \eta'(s) &= (1-2^{1-s})\zeta'(s)+2^{1-s} (\ln 2) \zeta(s) \\
                      &= -\frac{1}{2} \sum_{n=1}^\infty (-1)^{n-1}\left[\frac{\ln n}{n^s}-\frac{\ln (n+1)}{(n+1)^s}\right], \Re(s)>-1
\end{align}</math>
 
:<math>\begin{align}
  \Rightarrow \eta'(0) &= -\zeta'(0) - \ln 2 = -\frac{1}{2} \sum_{n=1}^\infty (-1)^{n-1}\left[\ln n-\ln (n+1)\right] \\
                        &= -\frac{1}{2} \sum_{n=1}^\infty (-1)^{n-1}\ln \frac{n}{n+1} \\
                        &= -\frac{1}{2} \left(\ln \frac{1}{2} - \ln \frac{2}{3} + \ln \frac{3}{4} - \ln \frac{4}{5} + \ln \frac{5}{6} - \cdots\right) \\
                        &=  \frac{1}{2} \left(\ln \frac{2}{1} + \ln \frac{2}{3} + \ln \frac{4}{3} + \ln \frac{4}{5} + \ln \frac{6}{5} + \cdots\right) \\
                        &=  \frac{1}{2} \ln\left(\frac{2}{1}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\cdot\cdots\right) = \frac{1}{2} \ln\frac{\pi}{2} \\
  \Rightarrow \zeta'(0) &= -\frac{1}{2} \ln\left(2 \pi\right)
\end{align}</math>
 
==See also==
*[[Viète's formula]], a different infinite product formula for π
*[[Leibniz formula for π]], an infinite sum that can be converted into an infinite [[Euler product]] for π
*[[Wallis sieve]]
 
==External links==
{{Reflist}}
 
[[Category:Articles containing proofs]]
[[Category:Pi algorithms]]

Revision as of 17:42, 2 March 2014

It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.



Here are some common dental emergencies:
Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.

At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.

Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.

Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.

Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.

Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.

Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.

In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.

For more information regarding dentist DC review our internet site.

Retrieved from "https://en.formulasearchengine.com/index.php?title=Two-port_network&oldid=237551"