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| {{About|Euler beta function}}
| | We all have an idea of what an essence body should look like, but in most cases, these imaginings can never be transferred to reality, no subject how hard we manoeuvre, adapt or change the physical body shape, there is a border to what can be achieved with cosmetics. In breast enlargement surgery, implants are used to increase the size of the breasts. Breast performance, health, and appearance are sensitive issues of great significance. Over 340,000 breast augmentations were performed in the United States in 2007, so leave your lovely girlfriend alone. It is must for you to check out, the work that surgeon had done earlier and have a count on the number of surgeries. <br><br>Using foods and supplements is proving to be a safer option than using surgery. The problem is graver if teardrop implants are used. If that doesn't pan out, you can ask your personal physician for a referral. Fenugreek is an herb with medicinal properties doing it a valuable ingredient in health-related and therapeutic products and solutions. Surgical breast implants can give you your choice of size relatively quickly, but breast implantation is an invasive procedure with risk for infections. <br><br>used to be the case that cheap breast augmentation prices were. Mostly, breast enlargement pills are used along with the creams, sprays and lotions. You must not fall in a pitfall while undergoing the surgery at a cheaper rate. Fenugreek aids in hormonal production, and facilitates the development of the mammary glands which 'feed' on estrogens. One of the most frequently performed cosmetic procedures in the India today. <br><br>Believe it or not, such mild workings within the human body tend to be of more benefit than pure concentrated forms of estrogen, especially for ladies who are approaching age 40 or slightly older. Moreover, for every honest practitioner who tried to help his patients, there were at least ten men looking to take advantage of others through scams and fakes. Though they confirm the company claims are misleading, these companies were not disposed to the same criterion as other creams produced from non-herbal ingredients. • Likely outcomes of the procedure and any risks or potential complications. Fennel also frequently found in some traditional weight loss ingredients. <br><br>Other specialists advocate placing the breast implant under the pectoral muscle in order to prevent interference with future mammograms. The circles you are making will become smaller as you reach the nipples. Don quai, one of the ingredients worn in the pills is a known carcinogen. There seems to be a fascination with the possibility of what life would be like with a fuller bust line. News that she feels more sexy now as more curves have been added on her. <br><br>Fenugreek and soya are easily available edible items that help in breast enlargement. The clear advantage to this method is that you can show the surgeon what you want in three-dimensions and on your own body. Pinks and light purples will go well for those with blue eyes or light skinned. Different brands of Breast Enlargement Pills consist of different ingredients. Some earlier mammography machines did tend to compress the breast tissue and are therefore best avoided.<br><br>When you loved this article and you would like to receive much more information relating to [http://www.lucky-house.info/sitemap/ bigger breast naturally] please visit the web site. |
| [[File:Beta function contour plot.png|thumb|[[Contour plot]] of the beta function]]
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| [[File:Beta function on real plane.png|thumb|A plot of the beta function for positive x and y values]]
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| In [[mathematics]], the '''beta function''', also called the [[Euler integral (disambiguation)|Euler integral]] of the first kind, is a [[special function]] defined by
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| :<math>
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| \mathrm{\Beta}(x,y) = \int_0^1t^{x-1}(1-t)^{y-1}\,\mathrm{d}t
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| \!</math>
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| for <math>\textrm{Re}(x), \textrm{Re}(y) > 0.\,</math>
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| <!-- The \, is to keep the formula rendered as PNG instead of HTML. Please don't remove it.-->
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| The beta function was studied by [[Leonhard Euler|Euler]] and [[Adrien-Marie Legendre|Legendre]] and was given its name by [[Jacques Philippe Marie Binet|Jacques Binet]]; its symbol Β is a [[Greek alphabet|Greek]] capital [[β]] rather than the similar [[Latin alphabet|Latin]] capital [[B]].
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| == Properties ==
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| The beta function is [[symmetric function|symmetric]], meaning that
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| :<math>
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| \Beta(x,y) = \Beta(y,x).
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| \!</math><ref name=Davis622>Davis (1972) 6.2.2 p.258</ref>
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| When x and y are positive integers, it follows from the definition of the [[gamma function]] <math>\Gamma\ </math> that:
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| :<math>
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| \Beta(x,y)=\dfrac{(x-1)!\,(y-1)!}{(x+y-1)!}
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| \!</math>
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| It has many other forms, including: | |
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| :<math>
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| \Beta(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)}
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| \!</math><ref name=Davis622/>
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| :<math>
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| \Beta(x,y) =
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| 2\int_0^{\pi/2}(\sin\theta)^{2x-1}(\cos\theta)^{2y-1}\,\mathrm{d}\theta,
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| \qquad \mathrm{Re}(x)>0,\ \mathrm{Re}(y)>0
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| \!</math><ref name=Davis621/>
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| :<math>
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| \Beta(x,y) =
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| \int_0^\infty\dfrac{t^{x-1}}{(1+t)^{x+y}}\,\mathrm{d}t,
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| \qquad \mathrm{Re}(x)>0,\ \mathrm{Re}(y)>0
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| \!</math><ref name=Davis621>Davis (1972) 6.2.1 p.258</ref>
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| :<math>
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| \Beta(x,y) =
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| \sum_{n=0}^\infty \dfrac{{n-y \choose n}} {x+n},
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| \!</math>
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| :<math>
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| \Beta(x,y) = \frac{x+y}{x y} \prod_{n=1}^\infty \left( 1+ \dfrac{x y}{n (x+y+n)}\right)^{-1},
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| \!</math>
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| The Beta function has several interesting properties, including
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| :<math>
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| \Beta(x,y) = \Beta(x, y+1) + \Beta(x+1, y)
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| \!</math>
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| :<math>
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| \Beta(x,y)\cdot(t \mapsto t_+^{x+y-1}) = (t \to t_+^{x-1}) * (t \to t_+^{y-1}) \qquad x\ge 1, y\ge 1,
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| \!</math>
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| :<math>
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| \Beta(x,y) \cdot \Beta(x+y,1-y) =
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| \dfrac{\pi}{x \sin(\pi y)},
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| \!</math>
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| <!-- :<math>
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| \Beta(x,y) =
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| \dfrac{1}{y}\sum_{n=0}^\infty(-1)^n\dfrac{y^{n+1}}{n!(x+n)}
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| \!</math> -->
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| where <math>t \mapsto t_+^x</math> is a [[truncated power function]] and the star denotes [[convolution]].
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| <!-- The \, is to keep the formula rendered as PNG instead of HTML. Please don't remove it.-->
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| The lowermost identity above shows in particular <math>\Gamma(\tfrac12) = \sqrt \pi</math>. Some of these identities, e.g. the trigonometric formula, can be applied to deriving the [[volume of an n-ball]] in [[Cartesian coordinates]].
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| Euler's integral for the beta function may be converted into an integral over the [[Pochhammer contour]] ''C'' as
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| :<math>\displaystyle (1-e^{2\pi i\alpha})(1-e^{2\pi i\beta})\Beta(\alpha,\beta) =\int_C t^{\alpha-1}(1-t)^{\beta-1} \, \mathrm{d}t.</math>
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| This Pochhammer contour integral converges for all values of ''α'' and ''β'' and so gives the [[analytic continuation]] of the beta function.
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| Just as the gamma function for integers describes [[factorial]]s, the beta function can define a [[binomial coefficient]] after adjusting indices:
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| :<math>{n \choose k} = \frac1{(n+1) \Beta(n-k+1, k+1)}.</math>
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| Moreover, for integer ''n'', <math>\Beta\,</math> can be integrated to give a closed form, an interpolation function for continuous values of ''k'':
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| :<math>{n \choose k} = (-1)^n n! \cfrac{\sin (\pi k)}{\pi \prod_{i=0}^n (k-i)}.</math>
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| The beta function was the first known [[S matrix|scattering amplitude]] in [[string theory]], first conjectured by [[Gabriele Veneziano]]. It also occurs in the theory of the [[preferential attachment]] process, a type of stochastic [[urn problem|urn process]].
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| == Relationship between gamma function and beta function ==
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| To derive the integral representation of the beta function, write the product of two factorials as
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| :<math>
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| \Gamma(x)\Gamma(y) =
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| \int_0^\infty\ e^{-u} u^{x-1}\,\mathrm{d}u \int_0^\infty\ e^{-v} v^{y-1}\,\mathrm{d}v
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| =\int_0^\infty\int_0^\infty\ e^{-u-v} u^{x-1}v^{y-1}\,\mathrm{d}u \,\mathrm{d}v.
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| \!</math>
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| Changing variables by putting ''u''=''zt'', ''v''=''z''(1-''t'')
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| shows that this is
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| :<math>
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| \int_{z=0}^\infty\int_{t=0}^1 e^{-z} (zt)^{x-1}(z(1-t))^{y-1}z\,\mathrm{d}z \,\mathrm{d}t
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| =\int_{z=0}^\infty e^{-z}z^{x+y-1} \,\mathrm{d}z\int_{t=0}^1t^{x-1}(1-t)^{y-1}\,\mathrm{d}t.
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| \!</math>
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| Hence
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| :<math>
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| \Gamma(x)\,\Gamma(y)=\Gamma(x+y)\Beta(x,y) .
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| </math>
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| The stated identity may be seen as a particular case of the identity for the [[convolution#Integration|integral of a convolution]]. Taking
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| :<math>f(u):=e^{-u} u^{x-1} 1_{\R_+}</math> and <math>g(u):=e^{-u} u^{y-1} 1_{\R_+}</math>, one has:
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| :<math>\Gamma(x)\Gamma(y)=\left(\int_{\R}f(u)\mathrm{d}u\right)\left(\int_{\R}g(u)\mathrm{d}u\right)=\int_{\R}(f*g)(u)\mathrm{d}u=\Beta(x, y)\,\Gamma(x+y)</math>.
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| == Derivatives ==
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| We have
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| :<math>{\partial \over \partial x} \mathrm{B}(x, y) = \mathrm{B}(x, y) \left( {\Gamma'(x) \over \Gamma(x)} - {\Gamma'(x + y) \over \Gamma(x + y)} \right) = \mathrm{B}(x, y) (\psi(x) - \psi(x + y)),</math>
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| where <math>\ \psi(x)</math> is the [[digamma function]].
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| == Integrals ==
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| The [[Nörlund–Rice integral]] is a contour integral involving the beta function.
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| ==Approximation==
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| [[Stirling's approximation]] gives the asymptotic formula
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| :<math>\Beta(x,y) \sim \sqrt {2\pi } \frac{{x^{x - \frac{1}{2}} y^{y - \frac{1}{2}} }}{{\left( {x + y} \right)^{x + y - \frac{1}{2}} }}</math>
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| for large ''x'' and large ''y''. If on the other hand ''x'' is large and ''y'' is fixed, then
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| :<math>\Beta(x,y) \sim \Gamma(y)\,x^{-y}.</math>
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| ==Incomplete beta function==
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| The '''incomplete beta function''', a generalization of the beta function, is defined as
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| :<math> \Beta(x;\,a,b) = \int_0^x t^{a-1}\,(1-t)^{b-1}\,\mathrm{d}t. \!</math>
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| For ''x'' = 1, the incomplete beta function coincides with the complete beta function. The relationship between the two functions is like that between the gamma function and its generalization the [[incomplete gamma function]].
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| The '''regularized incomplete beta function''' (or '''regularized beta function''' for short) is defined in terms of the incomplete beta function and the complete beta function:
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| :<math> I_x(a,b) = \dfrac{\Beta(x;\,a,b)}{\Beta(a,b)}. \!</math>
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| Working out the integral (one can use [[integration by parts]]) for integer values of ''a'' and ''b'', one finds:
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| :<math> I_x(a,b) = \sum_{j=a}^\infty \binom{a+b-1}{j} x^j (1-x)^{a+b-1-j}. </math>
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| The regularized incomplete beta function is the [[cumulative distribution function]] of the [[Beta distribution]], and is related to the [[cumulative distribution function]] of a [[random variable]] ''X'' from a [[binomial distribution]], where the "probability of success" is ''p'' and the sample size is ''n'':
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| :<math>F(k;n,p) = \Pr(X \le k) = I_{1-p}(n-k, k+1) = 1 - I_p(k+1,n-k). </math>
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| ===Properties===
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| <!-- (Many other properties could be listed here.)-->
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| :<math> I_0(a,b) = 0 \, </math>
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| :<math> I_1(a,b) = 1 \, </math>
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| :<math> I_x(a,1) = x^a \, </math>
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| :<math> I_x(a,b) = 1 - I_{1-x}(b,a) \, </math>
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| :<math> I_x(a+1,b) = I_x(a,b)-\frac{x^a(1-x)^b}{a B(a,b)} \, </math>.
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| ==Calculation==
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| Even if unavailable directly, the complete and incomplete beta function values can be calculated using functions commonly included in [[spreadsheet]] or [[computer algebra system]]s. With [[Microsoft Excel|Excel]] as an example, using the [[Gamma_function#Approximations|GammaLn]] and ([[cumulative distribution function|cumulative]]) [[beta distribution]] functions, we have:
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| :''Complete Beta Value = Exp(GammaLn(a) + GammaLn(b) − GammaLn(a + b))''
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| and, | |
| :''Incomplete Beta Value = BetaDist(x, a, b) * Exp(GammaLn(a) + GammaLn(b) − GammaLn(a + b))''.
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| These result from rearranging the formulae for the [[beta distribution]], and the incomplete beta and complete beta functions, which can also be defined as the ratio of the logs [[#Properties|as above]].
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| Similarly, in [[MATLAB]] and [[GNU Octave]], ''betainc'' (incomplete beta function), or in [[R_(programming_language)|R]], ''pbeta'' (probability of beta distribution) compute the [[Beta_distribution#Cumulative_distribution_function|regularized incomplete beta function]]—which is, in fact, the cumulative beta distribution—and so, to get the actual incomplete beta function, one must multiply the result of ''betainc'' by the result returned by the corresponding ''beta'' function.
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| ==See also==
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| * [[Beta distribution]]
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| * [[Binomial distribution]]
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| * [[Jacobi sum]], the analogue of the beta function over finite fields.
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| * [[Negative binomial distribution]]
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| * [[Yule–Simon distribution]]
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| * [[Uniform distribution (continuous)]]
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| * [[Gamma function]]
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| * [[Dirichlet distribution]]
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| {{No footnotes|date=November 2010}}
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| ==References==
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| {{reflist}}
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| * {{dlmf|authorlink=Richard Askey|first=R. A.|last= Askey|first2= R.|last2= Roy |id=5.12 }}
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| *{{citation | first1=M. | last1=Zelen | first2=N. C. | last2=Severo | chapter=26. Probability functions | pages=925-995 | editor1-last=Abramowitz | editor1-first=Milton | editor1-link=Milton Abramowitz | editor2-last=Stegun | editor2-first=Irene A. | editor2-link=Irene Stegun | title=[[Abramowitz and Stegun|Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables]] | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-61272-0 | year=1972}}
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| *{{citation | first=Philip J. | last=Davis | chapter=6. Gamma function and related functions | editor1-last=Abramowitz | editor1-first=Milton | editor1-link=Milton Abramowitz | editor2-last=Stegun | editor2-first=Irene A. | editor2-link=Irene Stegun | title=[[Abramowitz and Stegun|Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables]] | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-61272-0 | year=1972 | url=http://www.math.sfu.ca/~cbm/aands/page_258.htm }}
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| *{{dlmf|first=R. B. |last=Paris|id=8.17|title=Incomplete beta functions}}
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| * {{Citation | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 6.1 Gamma Function, Beta Function, Factorials | chapter-url=http://apps.nrbook.com/empanel/index.html?pg=256}}
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| ==External links==
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| * {{springer|title=Beta-function|id=p/b015960}}
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| * {{planetmath reference|id=6206|title=Evaluation of beta function using Laplace transform}}
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| * Arbitrarily accurate values can be obtained from:
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| ** [http://functions.wolfram.com The Wolfram Functions Site]: [http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=BetaRegularized Evaluate Beta Regularized Incomplete beta]
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| **danielsoper.com: [http://www.danielsoper.com/statcalc/calc36.aspx Incomplete Beta Function Calculator], [http://www.danielsoper.com/statcalc/calc37.aspx Regularized Incomplete Beta Function Calculator]
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| {{DEFAULTSORT:Beta Function}}
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| [[Category:Gamma and related functions]]
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| [[Category:Special hypergeometric functions]]
| |
We all have an idea of what an essence body should look like, but in most cases, these imaginings can never be transferred to reality, no subject how hard we manoeuvre, adapt or change the physical body shape, there is a border to what can be achieved with cosmetics. In breast enlargement surgery, implants are used to increase the size of the breasts. Breast performance, health, and appearance are sensitive issues of great significance. Over 340,000 breast augmentations were performed in the United States in 2007, so leave your lovely girlfriend alone. It is must for you to check out, the work that surgeon had done earlier and have a count on the number of surgeries.
Using foods and supplements is proving to be a safer option than using surgery. The problem is graver if teardrop implants are used. If that doesn't pan out, you can ask your personal physician for a referral. Fenugreek is an herb with medicinal properties doing it a valuable ingredient in health-related and therapeutic products and solutions. Surgical breast implants can give you your choice of size relatively quickly, but breast implantation is an invasive procedure with risk for infections.
used to be the case that cheap breast augmentation prices were. Mostly, breast enlargement pills are used along with the creams, sprays and lotions. You must not fall in a pitfall while undergoing the surgery at a cheaper rate. Fenugreek aids in hormonal production, and facilitates the development of the mammary glands which 'feed' on estrogens. One of the most frequently performed cosmetic procedures in the India today.
Believe it or not, such mild workings within the human body tend to be of more benefit than pure concentrated forms of estrogen, especially for ladies who are approaching age 40 or slightly older. Moreover, for every honest practitioner who tried to help his patients, there were at least ten men looking to take advantage of others through scams and fakes. Though they confirm the company claims are misleading, these companies were not disposed to the same criterion as other creams produced from non-herbal ingredients. • Likely outcomes of the procedure and any risks or potential complications. Fennel also frequently found in some traditional weight loss ingredients.
Other specialists advocate placing the breast implant under the pectoral muscle in order to prevent interference with future mammograms. The circles you are making will become smaller as you reach the nipples. Don quai, one of the ingredients worn in the pills is a known carcinogen. There seems to be a fascination with the possibility of what life would be like with a fuller bust line. News that she feels more sexy now as more curves have been added on her.
Fenugreek and soya are easily available edible items that help in breast enlargement. The clear advantage to this method is that you can show the surgeon what you want in three-dimensions and on your own body. Pinks and light purples will go well for those with blue eyes or light skinned. Different brands of Breast Enlargement Pills consist of different ingredients. Some earlier mammography machines did tend to compress the breast tissue and are therefore best avoided.
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