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{{DISPLAYTITLE: List of formulae involving {{pi}} }} | |||
{{Pi box}} | |||
The following is a list of significant formulae involving the [[mathematical constant]] [[pi|{{pi}}]]. The list contains only formulae whose significance is established either in the article on the formula itself, the article on pi, or the one on numerical [[approximations of π|approximations of {{pi}}]]. | |||
==Classical geometry== | |||
:<math>C = 2 \pi r = \pi d\!</math> | |||
where {{math|''C''}} is the circumference of a [[circle]], {{math|''r''}} is the radius and {{math|''d''}} is the diameter. | |||
:<math>A = \pi r^2\!</math> | |||
where {{math|A}} is the area of a circle and {{math|''r''}} is the radius. | |||
:<math>V = {4 \over 3}\pi r^3\!</math> | |||
where {{math|''V''}} is the volume of a [[sphere]] and {{math|''r''}} is the radius. | |||
:<math>SA = 4\pi r^2\!</math> | |||
where {{math|''SA''}} is the surface area of a sphere and {{math|''r''}} is the radius. | |||
==Analysis== | |||
===Integrals=== | |||
:<math>\int\limits_{-\infty}^{\infty} \text{sech}(x)dx = \pi \!</math> | |||
<br /> | |||
<math>\int\limits_{-\infty}^{\infty} \int\limits_{t}^{\infty} e^{-1/2t^2-x^2+xt} dxdt = \int\limits_{-\infty}^{\infty} \int\limits_{t}^{\infty} e^{^-t^2-1/2x^2+xt} dxdt = \pi\!</math> | |||
<br /> | |||
:<math>\int\limits_{-1}^1 \sqrt{1-x^2}\,dx = \frac{\pi}{2}\!</math> | |||
<br /> | |||
:<math>\int\limits_{-1}^1\frac{dx}{\sqrt{1-x^2}} = \pi\!</math> | |||
<br /> | |||
:<math>\int\limits_{-\infty}^\infty\frac{dx}{1+x^2} = \pi\!</math> (integral form of [[arctan]] over its entire domain, giving the period of [[Trigonometric_functions#tangent|tan]]). | |||
<br /> | |||
:<math>\int\limits_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}\!</math> (see [[gaussian integral]]). | |||
<br /> | |||
:<math>\oint\frac{dz}{z}=2\pi i\!</math> (when the path of integration winds once counterclockwise around 0. See also [[Cauchy's integral formula]]) | |||
<br /> | |||
:<math>\int\limits_{-\infty}^{\infty} \frac{\sin x}{x}\,dx=\pi \!</math> | |||
<br /> | |||
:<math>\int\limits_0^1 {x^4(1-x)^4 \over 1+x^2}\,dx = {22 \over 7} - \pi\!</math> (see also [[Proof that 22/7 exceeds π]]). | |||
<br /> | |||
===Efficient infinite series=== | |||
:<math>\sum_{k=0}^\infty\frac{k!}{(2k+1)!!}=\sum_{k=0}^\infty\frac{2^k k!^2}{(2k+1)!}=\frac{\pi}{2}\!</math> (see also [[double factorial]]) | |||
<br /> | |||
:<math>12 \sum^\infty_{k=0} \frac{(-1)^k (6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 640320^{3k + 3/2}}=\frac{1}{\pi}\!</math> (see [[Chudnovsky algorithm]]) | |||
<br /> | |||
:<math>\frac{2\sqrt{2}}{9801} \sum^\infty_{k=0} \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}=\frac{1}{\pi}\!</math> (see [[Srinivasa Ramanujan#Mathematical achievements|Srinivasa Ramanujan]]) | |||
<br /> | |||
:<math>\frac{\sqrt{3}}{6^5} \sum_{k = 0}^{\infty} \frac{((4k)!)^2(6k)!}{9^{k+1}(12k)!(2k)!} \left( \frac{127169}{12k + 1} - \frac{1070}{12k + 5} - \frac{131}{12k + 7} + \frac{2}{12k + 11}\right)=\pi\!</math><ref>Cetin Hakimoglu-Brown [http://iamned.com/math/infiniteseries.pdf Derivation of Rapidly Converging Infinite Series]</ref> | |||
<br /> | |||
The following are good for calculating arbitrary binary digits of {{pi}}: | |||
:<math>\sum_{k = 0}^{\infty} \frac{1}{16^k} \left( \frac{4}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6}\right)=\pi\!</math> (see [[Bailey-Borwein-Plouffe formula]]) | |||
<br /> | |||
:<math>\frac{1}{2^6} \sum_{n=0}^{\infty} \frac{{(-1)}^n}{2^{10n}} \left( - \frac{2^5}{4n+1} - \frac{1}{4n+3} + \frac{2^8}{10n+1} - \frac{2^6}{10n+3} - \frac{2^2}{10n+5} - \frac{2^2}{10n+7} + \frac{1}{10n+9} \right)=\pi\!</math> | |||
<br /> | |||
===Other infinite series=== | |||
:<math>\zeta(2) = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots = \frac{\pi^2}{6}\!</math> (see also [[Basel problem]] and [[Riemann zeta function]]) | |||
<br /> | |||
:<math>\zeta(4)= \frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \frac{1}{4^4} + \cdots = \frac{\pi^4}{90}\!</math> | |||
<br /> | |||
:<math>\zeta(2n) = \sum_{k=1}^{\infty} \frac{1}{k^{2n}}\, = \frac{1}{1^{2n}} + \frac{1}{2^{2n}} + \frac{1}{3^{2n}} + \frac{1}{4^{2n}} + \cdots = (-1)^{n+1}\frac{B_{2n}(2\pi)^{2n}}{2(2n)!}\!</math> , where ''B''<sub>2''n''</sub> is a [[Bernoulli number]]. | |||
<br /> | |||
:<math>\sum_{n=1}^{\infty} \frac{3^n - 1}{4^n}\, \zeta(n+1) = \pi\!</math><ref>[http://mathworld.wolfram.com/PiFormulas.html Weisstein, Eric W. "Pi Formulas", MathWorld]</ref> | |||
<br /> | |||
:<math>\sum_{n=0}^{\infty} {\left( \frac{(-1)^{n}}{2n+1} \right) }^1 = \frac{1}{1} - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \cdots = \arctan{1} = \frac{\pi}{4}\!</math> (see [[Leibniz formula for pi]]) | |||
<br /> | |||
:<math>\sum_{n=0}^{\infty} {\left( \frac{(-1)^{n}}{2n+1} \right) }^2 = \frac{1}{1^2} + \frac{1}{3^2} + \frac{1}{5^2} + \frac{1}{7^2} + \cdots = \frac{\pi^2}{8}\!</math> | |||
<br /> | |||
:<math>\sum_{n=0}^{\infty} {\left( \frac{(-1)^{n}}{2n+1} \right) }^3 = \frac{1}{1^3} - \frac{1}{3^3} + \frac{1}{5^3} - \frac{1}{7^3} + \cdots = \frac{\pi^3}{32}\!</math> | |||
<br /> | |||
:<math>\sum_{n=0}^{\infty} {\left( \frac{(-1)^{n}}{2n+1} \right) }^4 = \frac{1}{1^4} + \frac{1}{3^4} + \frac{1}{5^4} + \frac{1}{7^4} + \cdots = \frac{\pi^4}{96}\!</math> | |||
<br /> | |||
:<math>\sum_{n=0}^{\infty} {\left( \frac{(-1)^{n}}{2n+1} \right) }^5 = \frac{1}{1^5} - \frac{1}{3^5} + \frac{1}{5^5} - \frac{1}{7^5} + \cdots = \frac{5\pi^5}{1536}\!</math> | |||
<br /> | |||
:<math>\sum_{n=0}^{\infty} {\left( \frac{(-1)^{n}}{2n+1} \right) }^6 = \frac{1}{1^6} + \frac{1}{3^6} + \frac{1}{5^6} + \frac{1}{7^6} + \cdots = \frac{\pi^6}{960}\!</math> | |||
<br /> | |||
:<math> \frac{\pi}{4} = \frac{3}{4} \times \frac{5}{4} \times \frac{7}{8} \times \frac{11}{12} \times \frac{13}{12} \times \frac{17}{16} \times \frac{19}{20} \times \frac{23}{24} \times \frac{29}{28} \times \frac{31}{32} \times \cdots \! </math> ([[Leonhard Euler|Euler]]) | |||
:where the numerators are the odd primes; each denominator is the multiple of four nearest to the numerator. | |||
<br /> | |||
:<math> \pi = {{1}} + \frac{{1}}{{2}} + \frac{{1}}{{3}} + \frac{{1}}{{4}} - \frac{{1}}{{5}} + \frac{{1}}{{6}} + \frac{{1}}{{7}} + \frac{{1}}{{8}} + \frac{{1}}{{9}} - \frac{{1}}{{10}} + \frac{{1}}{{11}} + \frac{{1}}{{12}} - \frac{{1}}{{13}} + \cdots \!</math> (Euler, 1748) | |||
: After the first two terms, the signs are determined as follows: If the denominator is a prime of the form 4''m'' - 1, the sign is positive; if the denominator is a prime of the form 4''m'' + 1, the sign is negative; for composite numbers, the sign is equal the product of the signs of its factors.<ref>[[Carl B. Boyer]], ''A History of Mathematics'', Chapter 21., p. 488-489</ref> | |||
<br /> | |||
===Machin-like formulae=== | |||
See also [[Machin-like formula]]. | |||
: <math>\frac{\pi}{4} = 4 \arctan\frac{1}{5} - \arctan\frac{1}{239} \!</math> (the original [[John Machin|Machin's]] formula) | |||
<br /> | |||
:<math>\frac{\pi}{4} = \arctan 1</math> | |||
<br /> | |||
:<math>\frac{\pi}{4} = \arctan\frac{1}{2} + \arctan\frac{1}{3}\!</math> | |||
<br /> | |||
:<math>\frac{\pi}{4} = 2 \arctan\frac{1}{2} - \arctan\frac{1}{7}\!</math> | |||
<br /> | |||
:<math>\frac{\pi}{4} = 2 \arctan\frac{1}{3} + \arctan\frac{1}{7}\!</math> | |||
<br /> | |||
:<math>\frac{\pi}{4} = 5 \arctan\frac{1}{7} + 2 \arctan\frac{3}{79}\!</math> | |||
<br /> | |||
:<math>\frac{\pi}{4} = 12 \arctan\frac{1}{49} + 32 \arctan\frac{1}{57} - 5 \arctan\frac{1}{239} + 12 \arctan\frac{1}{110443}\!</math> | |||
<br /> | |||
:<math>\frac{\pi}{4} = 44 \arctan\frac{1}{57} + 7 \arctan\frac{1}{239} - 12 \arctan\frac{1}{682} + 24 \arctan\frac{1}{12943}\!</math> | |||
<br /> | |||
===Infinite series=== | |||
Some infinite series involving pi are:<ref>{{cite web |title=The world of Pi |url=http://www.pi314.net/eng/ramanujan.php |author= Simon Plouffe / David Bailey |publisher=Pi314.net |date= |accessdate=2011-01-29}}<br/>{{cite web|url=http://numbers.computation.free.fr/Constants/Pi/piSeries.html |title=Collection of series for {{pi}} |publisher=Numbers.computation.free.fr |date= |accessdate=2011-01-29}}</ref> | |||
{| class="wikitable" | |||
|- | |||
| <math>\pi=\frac{1}{Z}\!</math>|| <math>Z=\sum_{n=0}^{\infty } \frac{((2n)!)^3(42n+5)} {(n!)^6{16}^{3n+1}}\!</math> | |||
|- | |||
| <math>\pi=\frac{4}{Z}\!</math>|| <math>Z=\sum_{n=0}^{\infty } \frac{(-1)^n(4n)!(21460n+1123)} {(n!)^4{441}^{2n+1}{2}^{10n+1}}</math> | |||
|- | |||
| <math>\pi=\frac{4}{Z}\!</math>|| <math>Z=\sum_{n=0}^{\infty } \frac{(6n+1)\left ( \frac{1}{2} \right )^3_n} {{4^n}(n!)^3}\!</math> | |||
|- | |||
| <math>\pi=\frac{32}{Z}\!</math>|| <math>Z=\sum_{n=0}^{\infty } \left (\frac{\sqrt{5}-1}{2} \right )^{8n} \frac{(42n\sqrt{5} +30n + 5\sqrt{5}-1) \left ( \frac{1}{2} \right )^3_n} {{64^n}(n!)^3}\!</math> | |||
|- | |||
| <math>\pi=\frac{27}{4Z}\!</math>|| <math>Z=\sum_{n=0}^{\infty } \left (\frac{2}{27} \right )^n \frac{(15n+2)\left ( \frac{1}{2} \right )_n \left ( \frac{1}{3} \right )_n \left ( \frac{2}{3} \right )_n} {(n!)^3}\!</math> | |||
|- | |||
| <math>\pi=\frac{15\sqrt{3}}{2Z}\!</math>|| <math>Z=\sum_{n=0}^{\infty } \left ( \frac{4}{125} \right )^n \frac{(33n+4)\left ( \frac{1}{2} \right )_n \left ( \frac{1}{3} \right )_n \left ( \frac{2}{3} \right )_n} {(n!)^3}\!</math> | |||
|- | |||
| <math>\pi=\frac{85\sqrt{85}}{18\sqrt{3}Z}\!</math>|| <math>Z=\sum_{n=0}^{\infty } \left ( \frac{4}{85} \right )^n \frac{(133n+8)\left ( \frac{1}{2} \right )_n \left ( \frac{1}{6} \right )_n \left ( \frac{5}{6} \right )_n} {(n!)^3}\!</math> | |||
|- | |||
| <math>\pi=\frac{5\sqrt{5}}{2\sqrt{3}Z} \!</math>|| <math>Z=\sum_{n=0}^{\infty } \left ( \frac{4}{125} \right )^n \frac{(11n+1)\left ( \frac{1}{2} \right )_n \left ( \frac{1}{6} \right )_n \left ( \frac{5}{6} \right )_n} {(n!)^3}\!</math> | |||
|- | |||
| <math>\pi=\frac{2\sqrt{3}}{Z} \!</math>|| <math>Z=\sum_{n=0}^{\infty } \frac{(8n+1)\left ( \frac{1}{2} \right )_n \left ( \frac{1}{4} \right )_n \left ( \frac{3}{4} \right )_n} {(n!)^3{9}^{n}}\!</math> | |||
|- | |||
| <math>\pi=\frac{\sqrt{3}}{9Z} \!</math>|| <math>Z=\sum_{n=0}^{\infty } \frac{(40n+3)\left ( \frac{1}{2} \right )_n \left ( \frac{1}{4} \right )_n \left ( \frac{3}{4} \right )_n} {(n!)^3{49}^{2n+1}}\!</math> | |||
|- | |||
| <math>\pi=\frac{2\sqrt{11}}{11Z} \!</math>|| <math>Z=\sum_{n=0}^{\infty } \frac{(280n+19)\left ( \frac{1}{2} \right )_n \left ( \frac{1}{4} \right )_n \left ( \frac{3}{4} \right )_n} {(n!)^3{99}^{2n+1}}\!</math> | |||
|- | |||
| <math>\pi=\frac{\sqrt{2}}{4Z} \!</math>|| <math>Z=\sum_{n=0}^{\infty } \frac{(10n+1) \left ( \frac{1}{2} \right )_n \left ( \frac{1}{4} \right )_n \left ( \frac{3}{4} \right )_n} {(n!)^3{9}^{2n+1}}\!</math> | |||
|- | |||
| <math>\pi=\frac{4\sqrt{5}}{5Z} \!</math>|| <math>Z=\sum_{n=0}^{\infty } \frac{(644n+41) \left ( \frac{1}{2} \right )_n \left ( \frac{1}{4} \right )_n \left ( \frac{3}{4} \right )_n} {(n!)^35^n{72}^{2n+1}}\!</math> | |||
|- | |||
| <math>\pi=\frac{4\sqrt{3}}{3Z} \!</math>|| <math>Z=\sum_{n=0}^{\infty } \frac{(-1)^n(28n+3) \left ( \frac{1}{2} \right )_n \left ( \frac{1}{4} \right )_n \left ( \frac{3}{4} \right )_n} { (n!)^3{3^n}{4}^{n+1}}\!</math> | |||
|- | |||
| <math> \pi=\frac{4}{Z}\!</math>|| <math>Z=\sum_{n=0}^{\infty } \frac{(-1)^n(20n+3) \left ( \frac{1}{2} \right )_n \left ( \frac{1}{4} \right )_n \left ( \frac{3}{4} \right )_n} { (n!)^3{2}^{2n+1}}\!</math> | |||
|- | |||
| <math>\pi=\frac{72}{Z} \!</math>|| <math>Z=\sum_{n=0}^{\infty } \frac{(-1)^n(4n)!(260n+23)}{(n!)^44^{4n}18^{2n}}\!</math> | |||
|- | |||
| <math>\pi=\frac{3528}{Z} \!</math>|| <math>Z=\sum_{n=0}^{\infty } \frac{(-1)^n(4n)!(21460n+1123)}{(n!)^44^{4n}882^{2n}}\!</math> | |||
|} | |||
where | |||
: <math>(x)_n \!</math> | |||
is the [[Pochhammer symbol]] for the falling factorial. | |||
===Infinite products=== | |||
:<math> \prod_{n=1}^{\infty} \frac{4n^2}{4n^2-1} = \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{4}{3} \cdot \frac{16}{15} \cdot \frac{36}{35} \cdot \frac{64}{63} \cdots = \frac{\pi}{2} \!</math> (see also [[Wallis product]]) | |||
<br /> | |||
[[Franciscus Vieta|Vieta]]'s formula: | |||
:<math>\frac{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdot \cdots = \frac2\pi\!</math> | |||
=== Three continued fractions === | |||
:<math> | |||
\pi= {3 + \cfrac{1^2}{6 + \cfrac{3^2}{6 + \cfrac{5^2}{6 + \cfrac{7^2}{6 + \ddots\,}}}}} | |||
</math> | |||
<br /> | |||
:<math> | |||
\pi = \cfrac{4}{1 + \cfrac{1^2}{3 + \cfrac{2^2}{5 + \cfrac{3^2}{7 + \cfrac{4^2}{9 + \ddots}}}}} | |||
</math> | |||
<br /> | |||
:<math> | |||
\pi = \cfrac{4}{1 + \cfrac{1^2}{2 + \cfrac{3^2}{2 + \cfrac{5^2}{2 + \cfrac{7^2}{2 + \ddots}}}}}\, | |||
</math> | |||
For more on this third identity, see [[Euler's continued fraction formula]]. | |||
(See also [[continued fraction]] and [[generalized continued fraction]].) | |||
===Miscellaneous=== | |||
:<math>n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n\!</math> ([[Stirling's approximation]]) | |||
<br /> | |||
:<math>e^{i \pi} = -1\!</math> ([[Euler's identity]]) | |||
<br /> | |||
:<math>\sum_{k=1}^{n} \varphi (k) \sim \frac{3n^2}{\pi^2}\!</math> (see [[Euler's totient function]]) | |||
<br /> | |||
:<math>\sum_{k=1}^{n} \frac {\varphi (k)} {k} \sim \frac{6n}{\pi^2}\!</math> (see [[Euler's totient function]]) | |||
<br /> | |||
:<math>\Gamma\left({1 \over 2}\right)=\sqrt{\pi}\!</math> (see also [[gamma function]]) | |||
<br /> | |||
:<math>\pi = \frac{\Gamma\left({1/4}\right)^{4/3} \mathrm{agm}(1, \sqrt{2})^{2/3}}{2}\!</math> (where agm is the [[arithmetic-geometric mean]]) | |||
<br /> | |||
:<math>\lim_{n\rightarrow \infty}\frac{1}{n^2} \sum_{k=1}^n (n\;\bmod\;k) = 1-\frac{\pi^2}{12}\!</math> (where mod is the modulo function which gives the rest of a division this formula is getting better for higher n) | |||
<br /> | |||
:<math> \pi = \lim_{n \rightarrow \infty} \frac{4}{n^2} \sum_{k=1}^n \sqrt{n^2 - k^2} </math> ([[Riemann sum]] to evaluate the area of the unit circle) | |||
<br /> | |||
:<math> \pi = \lim_{n \rightarrow \infty} \frac{2^{4n}}{n {2n\choose n}^2} </math> (by [[Stirling's approximation]]) | |||
<br /> | |||
==Physics== | |||
*The [[cosmological constant]]: | |||
::<math>\Lambda = {{8\pi G} \over {3c^2}} \rho\!</math> | |||
<br /> | |||
*[[Uncertainty principle|Heisenberg's uncertainty principle]]: | |||
::<math> \Delta x\, \Delta p \ge \frac{h}{4\pi} \!</math> | |||
<br /> | |||
*[[Einstein's field equation]] of [[general relativity]]: | |||
::<math> R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R + \Lambda g_{\mu\nu} = {8 \pi G \over c^4} T_{\mu\nu} \!</math> | |||
<br /> | |||
*[[Coulomb's law]] for the [[electric force]]: | |||
::<math> F = \frac{\left|q_1q_2\right|}{4 \pi \varepsilon_0 r^2}\!</math> | |||
<br /> | |||
*[[Permeability (electromagnetism)|Magnetic permeability of free space]]: | |||
::<math> \mu_0 = 4 \pi \cdot 10^{-7}\,\mathrm{N/A^2}\!</math> | |||
<br /> | |||
*Period of a simple [[pendulum]] with small amplitude | |||
:::<math>T \approx 2\pi \sqrt\frac{L}{g}\!</math> | |||
==See also== | |||
* [[Pi]] | |||
* [[List of topics related to π]] | |||
==References== | |||
{{reflist|2}} | |||
==Further reading== | |||
* Peter Borwein, ''[http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P159.pdf The Amazing Number Pi]'' | |||
* Kazuya Kato, Nobushige Kurokawa, Saito Takeshi: ''Number Theory 1: Fermat's Dream.'' American Mathematical Society, Providence 1993, ISBN 0-8218-0863-X. | |||
{{DEFAULTSORT:Formulae Involving Pi, List Of}} | |||
[[Category:Pi]] | |||
[[Category:Mathematics-related lists|Pi]] | |||
[[Category:Pi algorithms]] | |||
Revision as of 01:29, 9 January 2013
Template:Pi box The following is a list of significant formulae involving the mathematical constant Potter or Ceramic Artist Harry Rave from Cobden, spends time with hobbies for instance magic, property developers [https://www.manateegalleries.com/?option=com_k2&view=itemlist&task=user&id=30857 house in singapore] singapore and fitness. Finds inspiration through travel and just spent 7 months at Keoladeo National Park.. The list contains only formulae whose significance is established either in the article on the formula itself, the article on pi, or the one on numerical approximations of Potter or Ceramic Artist Harry Rave from Cobden, spends time with hobbies for instance magic, property developers [https://www.manateegalleries.com/?option=com_k2&view=itemlist&task=user&id=30857 house in singapore] singapore and fitness. Finds inspiration through travel and just spent 7 months at Keoladeo National Park..
Classical geometry
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If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. is the circumference of a circle, Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. is the radius and Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. is the diameter.
where Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. is the area of a circle and Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. is the radius.
where Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. is the volume of a sphere and Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. is the radius.
where Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. is the surface area of a sphere and Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.
Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.
There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity
This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.
A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.
If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.
Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. is the radius.
Analysis
Integrals
- (see gaussian integral).
- (when the path of integration winds once counterclockwise around 0. See also Cauchy's integral formula)
- (see also Proof that 22/7 exceeds π).
Efficient infinite series
- (see also double factorial)
- (see Chudnovsky algorithm)
- (see Srinivasa Ramanujan)
The following are good for calculating arbitrary binary digits of Potter or Ceramic Artist Harry Rave from Cobden, spends time with hobbies for instance magic, property developers house in singapore singapore and fitness. Finds inspiration through travel and just spent 7 months at Keoladeo National Park.:
Other infinite series
- (see also Basel problem and Riemann zeta function)
- , where B2n is a Bernoulli number.
- (see Leibniz formula for pi)
- (Euler)
- where the numerators are the odd primes; each denominator is the multiple of four nearest to the numerator.
- After the first two terms, the signs are determined as follows: If the denominator is a prime of the form 4m - 1, the sign is positive; if the denominator is a prime of the form 4m + 1, the sign is negative; for composite numbers, the sign is equal the product of the signs of its factors.[3]
Machin-like formulae
See also Machin-like formula.
- (the original Machin's formula)
Infinite series
Some infinite series involving pi are:[4]
where
is the Pochhammer symbol for the falling factorial.
Infinite products
- (see also Wallis product)
Vieta's formula:
Three continued fractions
For more on this third identity, see Euler's continued fraction formula.
(See also continued fraction and generalized continued fraction.)
Miscellaneous
- (see Euler's totient function)
- (see Euler's totient function)
- (see also gamma function)
- (where agm is the arithmetic-geometric mean)
- (where mod is the modulo function which gives the rest of a division this formula is getting better for higher n)
- (Riemann sum to evaluate the area of the unit circle)
Physics
- Coulomb's law for the electric force:
- Period of a simple pendulum with small amplitude
See also
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.
Further reading
- Peter Borwein, The Amazing Number Pi
- Kazuya Kato, Nobushige Kurokawa, Saito Takeshi: Number Theory 1: Fermat's Dream. American Mathematical Society, Providence 1993, ISBN 0-8218-0863-X.
- ↑ Cetin Hakimoglu-Brown Derivation of Rapidly Converging Infinite Series
- ↑ Weisstein, Eric W. "Pi Formulas", MathWorld
- ↑ Carl B. Boyer, A History of Mathematics, Chapter 21., p. 488-489
- ↑ Template:Cite web
Template:Cite web