Trigonometric integral

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Contour plot of the beta function

A plot of the beta function for positive x and y values

In mathematics, the beta function, also called the Euler integral of the first kind, is a special function defined by

$\mathrm {\mathrm {B} } (x,y)=\int _{0}^{1}t^{x-1}(1-t)^{y-1}\,\mathrm {d} t\!$

for ${\textrm {Re}}(x),{\textrm {Re}}(y)>0.\,$

The beta function was studied by Euler and Legendre and was given its name by Jacques Binet; its symbol Β is a Greek capital β rather than the similar Latin capital B.

Properties

The beta function is symmetric, meaning that

$\mathrm {B} (x,y)=\mathrm {B} (y,x).\!$

When x and y are positive integers, it follows from the definition of the gamma function $\Gamma \$ that:

$\mathrm {B} (x,y)={\dfrac {(x-1)!\,(y-1)!}{(x+y-1)!}}\!$

It has many other forms, including:

$\mathrm {B} (x,y)={\dfrac {\Gamma (x)\,\Gamma (y)}{\Gamma (x+y)}}\!$

$\mathrm {B} (x,y)=2\int _{0}^{\pi /2}(\sin \theta )^{2x-1}(\cos \theta )^{2y-1}\,\mathrm {d} \theta ,\qquad \mathrm {Re} (x)>0,\ \mathrm {Re} (y)>0\!$

$\mathrm {B} (x,y)=\int _{0}^{\infty }{\dfrac {t^{x-1}}{(1+t)^{x+y}}}\,\mathrm {d} t,\qquad \mathrm {Re} (x)>0,\ \mathrm {Re} (y)>0\!$

$\mathrm {B} (x,y)=\sum _{n=0}^{\infty }{\dfrac {n-y \choose n}{x+n}},\!$

$\mathrm {B} (x,y)={\frac {x+y}{xy}}\prod _{n=1}^{\infty }\left(1+{\dfrac {xy}{n(x+y+n)}}\right)^{-1},\!$

The Beta function has several interesting properties, including

$\mathrm {B} (x,y)=\mathrm {B} (x,y+1)+\mathrm {B} (x+1,y)\!$

$\mathrm {B} (x,y)\cdot (t\mapsto t_{+}^{x+y-1})=(t\to t_{+}^{x-1})*(t\to t_{+}^{y-1})\qquad x\geq 1,y\geq 1,\!$

$\mathrm {B} (x,y)\cdot \mathrm {B} (x+y,1-y)={\dfrac {\pi }{x\sin(\pi y)}},\!$

where $t\mapsto t_{+}^{x}$ is a truncated power function and the star denotes convolution. The lowermost identity above shows in particular $\Gamma ({\tfrac {1}{2}})={\sqrt {\pi }}$. Some of these identities, e.g. the trigonometric formula, can be applied to deriving the volume of an n-ball in Cartesian coordinates.

Euler's integral for the beta function may be converted into an integral over the Pochhammer contour C as

$\displaystyle (1-e^{2\pi i\alpha })(1-e^{2\pi i\beta })\mathrm {B} (\alpha ,\beta )=\int _{C}t^{\alpha -1}(1-t)^{\beta -1}\,\mathrm {d} t.$

This Pochhammer contour integral converges for all values of α and β and so gives the analytic continuation of the beta function.

Just as the gamma function for integers describes factorials, the beta function can define a binomial coefficient after adjusting indices:

${n \choose k}={\frac {1}{(n+1)\mathrm {B} (n-k+1,k+1)}}.$

Moreover, for integer n, $\mathrm {B} \,$ can be integrated to give a closed form, an interpolation function for continuous values of k:

${n \choose k}=(-1)^{n}n!{\cfrac {\sin(\pi k)}{\pi \prod _{i=0}^{n}(k-i)}}.$

The beta function was the first known 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 process.

Relationship between gamma function and beta function

To derive the integral representation of the beta function, write the product of two factorials as

$\Gamma (x)\Gamma (y)=\int _{0}^{\infty }\ e^{-u}u^{x-1}\,\mathrm {d} u\int _{0}^{\infty }\ e^{-v}v^{y-1}\,\mathrm {d} v=\int _{0}^{\infty }\int _{0}^{\infty }\ e^{-u-v}u^{x-1}v^{y-1}\,\mathrm {d} u\,\mathrm {d} v.\!$

Changing variables by putting u=zt, v=z(1-t) shows that this is

$\int _{z=0}^{\infty }\int _{t=0}^{1}e^{-z}(zt)^{x-1}(z(1-t))^{y-1}z\,\mathrm {d} z\,\mathrm {d} t=\int _{z=0}^{\infty }e^{-z}z^{x+y-1}\,\mathrm {d} z\int _{t=0}^{1}t^{x-1}(1-t)^{y-1}\,\mathrm {d} t.\!$

Hence

$\Gamma (x)\,\Gamma (y)=\Gamma (x+y)\mathrm {B} (x,y).$

The stated identity may be seen as a particular case of the identity for the integral of a convolution. Taking

$f(u):=e^{-u}u^{x-1}1_{\mathbb {R} _{+}}$ and $g(u):=e^{-u}u^{y-1}1_{\mathbb {R} _{+}}$, one has:

$\Gamma (x)\Gamma (y)=\left(\int _{\mathbb {R} }f(u)\mathrm {d} u\right)\left(\int _{\mathbb {R} }g(u)\mathrm {d} u\right)=\int _{\mathbb {R} }(f*g)(u)\mathrm {d} u=\mathrm {B} (x,y)\,\Gamma (x+y)$.

Derivatives

We have

${\partial \over \partial x}\mathrm {B} (x,y)=\mathrm {B} (x,y)\left({\Gamma '(x) \over \Gamma (x)}-{\Gamma '(x+y) \over \Gamma (x+y)}\right)=\mathrm {B} (x,y)(\psi (x)-\psi (x+y)),$

where $\ \psi (x)$ is the digamma function.

Integrals

The Nörlund–Rice integral is a contour integral involving the beta function.

Approximation

Stirling's approximation gives the asymptotic formula

$\mathrm {B} (x,y)\sim {\sqrt {2\pi }}{\frac {x^{x-{\frac {1}{2}}}y^{y-{\frac {1}{2}}}}{\left({x+y}\right)^{x+y-{\frac {1}{2}}}}}$

for large x and large y. If on the other hand x is large and y is fixed, then

$\mathrm {B} (x,y)\sim \Gamma (y)\,x^{-y}.$

Incomplete beta function

The incomplete beta function, a generalization of the beta function, is defined as

$\mathrm {B} (x;\,a,b)=\int _{0}^{x}t^{a-1}\,(1-t)^{b-1}\,\mathrm {d} t.\!$

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.

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:

$I_{x}(a,b)={\dfrac {\mathrm {B} (x;\,a,b)}{\mathrm {B} (a,b)}}.\!$

Working out the integral (one can use integration by parts) for integer values of a and b, one finds:

$I_{x}(a,b)=\sum _{j=a}^{\infty }{\binom {a+b-1}{j}}x^{j}(1-x)^{a+b-1-j}.$

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:

$F(k;n,p)=\Pr(X\leq k)=I_{1-p}(n-k,k+1)=1-I_{p}(k+1,n-k).$

Properties

$I_{0}(a,b)=0\,$

$I_{1}(a,b)=1\,$

$I_{x}(a,1)=x^{a}\,$

$I_{x}(a,b)=1-I_{1-x}(b,a)\,$

$I_{x}(a+1,b)=I_{x}(a,b)-{\frac {x^{a}(1-x)^{b}}{aB(a,b)}}\,$.

Calculation

Even if unavailable directly, the complete and incomplete beta function values can be calculated using functions commonly included in spreadsheet or computer algebra systems. With Excel as an example, using the GammaLn and (cumulative) beta distribution functions, we have:

Complete Beta Value = Exp(GammaLn(a) + GammaLn(b) − GammaLn(a + b))

and,

Incomplete Beta Value = BetaDist(x, a, b) * Exp(GammaLn(a) + GammaLn(b) − GammaLn(a + b)).

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 as above.

Similarly, in MATLAB and GNU Octave, betainc (incomplete beta function), or in R, pbeta (probability of beta distribution) compute the 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.

See also

• Beta distribution
• Binomial distribution
• Jacobi sum, the analogue of the beta function over finite fields.
• Negative binomial distribution
• Yule–Simon distribution
• Uniform distribution (continuous)
• Gamma function
• Dirichlet distribution

References

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  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Template:Dlmf
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

External links

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• Arbitrarily accurate values can be obtained from:
  • The Wolfram Functions Site: Evaluate Beta Regularized Incomplete beta
  • danielsoper.com: Incomplete Beta Function Calculator, Regularized Incomplete Beta Function Calculator