Beta function




This article is about the Euler beta function. For other uses, see Beta function (disambiguation).




Contour plot of the beta function


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


B(x,y)=∫01tx−1(1−t)y−1dt{displaystyle mathrm {B} (x,y)=int _{0}^{1}t^{x-1}(1-t)^{y-1},dt}{displaystyle mathrm {B} (x,y)=int _{0}^{1}t^{x-1}(1-t)^{y-1},dt}

for Re x > 0, 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 beta rather than the similar Latin capital B or the Greek lowercase β.




Contents






  • 1 Properties


  • 2 Relationship between gamma function and beta function


  • 3 Derivatives


  • 4 Integrals


  • 5 Approximation


  • 6 Incomplete beta function


    • 6.1 Properties




  • 7 Multivariate beta function


  • 8 Software implementation


  • 9 See also


  • 10 References


  • 11 External links





Properties


The beta function is symmetric, meaning that[1]


B(x,y)=B(y,x).{displaystyle mathrm {B} (x,y)=mathrm {B} (y,x).}{displaystyle mathrm {B} (x,y)=mathrm {B} (y,x).}

A key property of the Beta function is its relationship to the gamma function; proof is given below in the section on relationship between gamma function and beta function[1]


B(x,y)=Γ(x)Γ(y)Γ(x+y).{displaystyle mathrm {B} (x,y)={frac {Gamma (x),Gamma (y)}{Gamma (x+y)}}.}{displaystyle mathrm {B} (x,y)={frac {Gamma (x),Gamma (y)}{Gamma (x+y)}}.}

When x and y are positive integers, it follows from the definition of the gamma function Γ that:[2]


B(x,y)=(x−1)!(y−1)!(x+y−1)!B(x,y)=2∫0π/2(sin⁡θ)2x−1(cos⁡θ)2y−1dθ,Re⁡(x)>0, Re⁡(y)>0B(x,y)=∫0∞tx−1(1+t)x+ydt,Re⁡(x)>0, Re⁡(y)>0B(x,y)=n∫01tnx−1(1−tn)y−1dt,Re⁡(x)>0, Re⁡(y)>0, n>0B(x,y)=∑n=0∞(n−yn)x+n,B(x,y)=x+yxy∏n=1∞(1+xyn(x+y+n))−1,{displaystyle {begin{aligned}mathrm {B} (x,y)&={dfrac {(x-1)!,(y-1)!}{(x+y-1)!}}\[6pt]mathrm {B} (x,y)&=2int _{0}^{pi /2}(sin theta )^{2x-1}(cos theta )^{2y-1},dtheta ,&&operatorname {Re} (x)>0, operatorname {Re} (y)>0\[6pt]mathrm {B} (x,y)&=int _{0}^{infty }{frac {t^{x-1}}{(1+t)^{x+y}}},dt,&&operatorname {Re} (x)>0, operatorname {Re} (y)>0\[6pt]mathrm {B} (x,y)&=nint _{0}^{1}t^{nx-1}(1-t^{n})^{y-1},dt,&&operatorname {Re} (x)>0, operatorname {Re} (y)>0, n>0\[6pt]mathrm {B} (x,y)&=sum _{n=0}^{infty }{frac {binom {n-y}{n}}{x+n}},\[6pt]mathrm {B} (x,y)&={frac {x+y}{xy}}prod _{n=1}^{infty }left(1+{dfrac {xy}{n(x+y+n)}}right)^{-1},end{aligned}}}{displaystyle {begin{aligned}mathrm {B} (x,y)&={dfrac {(x-1)!,(y-1)!}{(x+y-1)!}}\[6pt]mathrm {B} (x,y)&=2int _{0}^{pi /2}(sin theta )^{2x-1}(cos theta )^{2y-1},dtheta ,&&operatorname {Re} (x)>0, operatorname {Re} (y)>0\[6pt]mathrm {B} (x,y)&=int _{0}^{infty }{frac {t^{x-1}}{(1+t)^{x+y}}},dt,&&operatorname {Re} (x)>0, operatorname {Re} (y)>0\[6pt]mathrm {B} (x,y)&=nint _{0}^{1}t^{nx-1}(1-t^{n})^{y-1},dt,&&operatorname {Re} (x)>0, operatorname {Re} (y)>0, n>0\[6pt]mathrm {B} (x,y)&=sum _{n=0}^{infty }{frac {binom {n-y}{n}}{x+n}},\[6pt]mathrm {B} (x,y)&={frac {x+y}{xy}}prod _{n=1}^{infty }left(1+{dfrac {xy}{n(x+y+n)}}right)^{-1},end{aligned}}}

The Beta function satisfies several interesting identities, including


B(x,y)=B(x,y+1)+B(x+1,y)B(x+1,y)=B(x,y)⋅xx+yB(x,y+1)=B(x,y)⋅yx+yB(x,y)⋅(t↦t+x+y−1)=(t→t+x−1)∗(t→t+y−1)x≥1,y≥1,B(x,y)⋅B(x+y,1−y)=πxsin⁡(πy)B(x,1−x)=πsin⁡(πx)B(1,x)=1x{displaystyle {begin{aligned}mathrm {B} (x,y)&=mathrm {B} (x,y+1)+mathrm {B} (x+1,y)\[6pt]mathrm {B} (x+1,y)&=mathrm {B} (x,y)cdot {dfrac {x}{x+y}}\[6pt]mathrm {B} (x,y+1)&=mathrm {B} (x,y)cdot {dfrac {y}{x+y}}\[6pt]mathrm {B} (x,y)&cdot left(tmapsto t_{+}^{x+y-1}right)={Big (}tto t_{+}^{x-1}{Big )}*{Big (}tto t_{+}^{y-1}{Big )}&&xgeq 1,ygeq 1,\[6pt]mathrm {B} (x,y)&cdot mathrm {B} (x+y,1-y)={frac {pi }{xsin(pi y)}}\[6pt]mathrm {B} (x,1-x)&={dfrac {pi }{sin(pi x)}}\[6pt]mathrm {B} (1,x)&={dfrac {1}{x}}end{aligned}}}{displaystyle {begin{aligned}mathrm {B} (x,y)&=mathrm {B} (x,y+1)+mathrm {B} (x+1,y)\[6pt]mathrm {B} (x+1,y)&=mathrm {B} (x,y)cdot {dfrac {x}{x+y}}\[6pt]mathrm {B} (x,y+1)&=mathrm {B} (x,y)cdot {dfrac {y}{x+y}}\[6pt]mathrm {B} (x,y)&cdot left(tmapsto t_{+}^{x+y-1}right)={Big (}tto t_{+}^{x-1}{Big )}*{Big (}tto t_{+}^{y-1}{Big )}&&xgeq 1,ygeq 1,\[6pt]mathrm {B} (x,y)&cdot mathrm {B} (x+y,1-y)={frac {pi }{xsin(pi y)}}\[6pt]mathrm {B} (x,1-x)&={dfrac {pi }{sin(pi x)}}\[6pt]mathrm {B} (1,x)&={dfrac {1}{x}}end{aligned}}}

where t ↦ tx
+ is a truncated power function and the star denotes convolution.


The lowermost identity above shows in particular Γ(1 / 2) = √π. 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


(1−e2πiα)(1−e2πiβ)B(α,β)=∫Ctα−1(1−t)β−1dt.{displaystyle left(1-e^{2pi ialpha }right)left(1-e^{2pi ibeta }right)mathrm {B} (alpha ,beta )=int _{C}t^{alpha -1}(1-t)^{beta -1},dt.}{displaystyle left(1-e^{2pi ialpha }right)left(1-e^{2pi ibeta }right)mathrm {B} (alpha ,beta )=int _{C}t^{alpha -1}(1-t)^{beta -1},dt.}

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:


(nk)=1(n+1)B(n−k+1,k+1).{displaystyle {binom {n}{k}}={frac {1}{(n+1)mathrm {B} (n-k+1,k+1)}}.}{displaystyle {binom {n}{k}}={frac {1}{(n+1)mathrm {B} (n-k+1,k+1)}}.}

Moreover, for integer n, Β can be factored to give a closed form, an interpolation function for continuous values of k:


(nk)=(−1)nn!⋅sin⁡(πk)π∏i=0n(k−i).{displaystyle {binom {n}{k}}=(-1)^{n},n!cdot {frac {sin(pi k)}{pi displaystyle prod _{i=0}^{n}(k-i)}}.}{displaystyle {binom {n}{k}}=(-1)^{n},n!cdot {frac {sin(pi k)}{pi displaystyle 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


A simple derivation of the relation can be found in Emil Artin's book The Gamma Function, page 18–19.[3]


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


Γ(x)Γ(y)=∫u=0∞ e−uux−1du⋅∫v=0∞ e−vvy−1dv=∫v=0∞∫u=0∞ e−u−vux−1vy−1dudv.{displaystyle {begin{aligned}Gamma (x)Gamma (y)&=int _{u=0}^{infty } e^{-u}u^{x-1},ducdot int _{v=0}^{infty } e^{-v}v^{y-1},dv\[6pt]&=int _{v=0}^{infty }int _{u=0}^{infty } e^{-u-v}u^{x-1}v^{y-1},du,dv.end{aligned}}}{displaystyle {begin{aligned}Gamma (x)Gamma (y)&=int _{u=0}^{infty } e^{-u}u^{x-1},ducdot int _{v=0}^{infty } e^{-v}v^{y-1},dv\[6pt]&=int _{v=0}^{infty }int _{u=0}^{infty } e^{-u-v}u^{x-1}v^{y-1},du,dv.end{aligned}}}

Changing variables by u = f(z,t) = zt and v = g(z,t) = z(1 − t) shows that this is


Γ(x)Γ(y)=∫z=0∞∫t=01e−z(zt)x−1(z(1−t))y−1|J(z,t)|dtdz=∫z=0∞∫t=01e−z(zt)x−1(z(1−t))y−1zdtdz=∫z=0∞e−zzx+y−1dz⋅∫t=01tx−1(1−t)y−1dt=Γ(x+y)B(x,y),{displaystyle {begin{aligned}Gamma (x)Gamma (y)&=int _{z=0}^{infty }int _{t=0}^{1}e^{-z}(zt)^{x-1}(z(1-t))^{y-1}{big |}J(z,t){big |},dt,dz\[6pt]&=int _{z=0}^{infty }int _{t=0}^{1}e^{-z}(zt)^{x-1}(z(1-t))^{y-1}z,dt,dz\[6pt]&=int _{z=0}^{infty }e^{-z}z^{x+y-1},dzcdot int _{t=0}^{1}t^{x-1}(1-t)^{y-1},dt\&=Gamma (x+y),mathrm {B} (x,y),end{aligned}}}{displaystyle {begin{aligned}Gamma (x)Gamma (y)&=int _{z=0}^{infty }int _{t=0}^{1}e^{-z}(zt)^{x-1}(z(1-t))^{y-1}{big |}J(z,t){big |},dt,dz\[6pt]&=int _{z=0}^{infty }int _{t=0}^{1}e^{-z}(zt)^{x-1}(z(1-t))^{y-1}z,dt,dz\[6pt]&=int _{z=0}^{infty }e^{-z}z^{x+y-1},dzcdot int _{t=0}^{1}t^{x-1}(1-t)^{y-1},dt\&=Gamma (x+y),mathrm {B} (x,y),end{aligned}}}

where |J(z,t)| is the absolute value of the Jacobian determinant of u = f(z,t) and v = g(z,t).


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


f(u):=e−uux−11R+g(u):=e−uuy−11R+,{displaystyle {begin{aligned}f(u)&:=e^{-u}u^{x-1}1_{mathbb {R} _{+}}\g(u)&:=e^{-u}u^{y-1}1_{mathbb {R} _{+}},end{aligned}}}{displaystyle {begin{aligned}f(u)&:=e^{-u}u^{x-1}1_{mathbb {R} _{+}}\g(u)&:=e^{-u}u^{y-1}1_{mathbb {R} _{+}},end{aligned}}}

one has:


Γ(x)Γ(y)=∫Rf(u)du⋅∫Rg(u)du=∫R(f∗g)(u)du=B(x,y)Γ(x+y).{displaystyle Gamma (x)Gamma (y)=int _{mathbb {R} }f(u),ducdot int _{mathbb {R} }g(u),du=int _{mathbb {R} }(f*g)(u),du=mathrm {B} (x,y),Gamma (x+y).}{displaystyle Gamma (x)Gamma (y)=int _{mathbb {R} }f(u),ducdot int _{mathbb {R} }g(u),du=int _{mathbb {R} }(f*g)(u),du=mathrm {B} (x,y),Gamma (x+y).}


Derivatives


We have


∂∂xB(x,y)=B(x,y)(Γ′(x)Γ(x)−Γ′(x+y)Γ(x+y))=B(x,y)(ψ(x)−ψ(x+y)),{displaystyle {frac {partial }{partial x}}mathrm {B} (x,y)=mathrm {B} (x,y)left({frac {Gamma '(x)}{Gamma (x)}}-{frac {Gamma '(x+y)}{Gamma (x+y)}}right)=mathrm {B} (x,y){big (}psi (x)-psi (x+y){big )},}{displaystyle {frac {partial }{partial x}}mathrm {B} (x,y)=mathrm {B} (x,y)left({frac {Gamma '(x)}{Gamma (x)}}-{frac {Gamma '(x+y)}{Gamma (x+y)}}right)=mathrm {B} (x,y){big (}psi (x)-psi (x+y){big )},}

where ψ(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


B(x,y)∼2πxx−1/2yy−1/2(x+y)x+y−1/2{displaystyle mathrm {B} (x,y)sim {sqrt {2pi }}{frac {x^{x-1/2}y^{y-1/2}}{({x+y})^{x+y-1/2}}}}{displaystyle mathrm {B} (x,y)sim {sqrt {2pi }}{frac {x^{x-1/2}y^{y-1/2}}{({x+y})^{x+y-1/2}}}}

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


B(x,y)∼Γ(y)x−y.{displaystyle mathrm {B} (x,y)sim Gamma (y),x^{-y}.}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


B(x;a,b)=∫0xta−1(1−t)b−1dt.{displaystyle mathrm {B} (x;,a,b)=int _{0}^{x}t^{a-1},(1-t)^{b-1},dt.}{displaystyle mathrm {B} (x;,a,b)=int _{0}^{x}t^{a-1},(1-t)^{b-1},dt.}

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:


Ix(a,b)=B(x;a,b)B(a,b).{displaystyle I_{x}(a,b)={frac {mathrm {B} (x;,a,b)}{mathrm {B} (a,b)}}.}{displaystyle I_{x}(a,b)={frac {mathrm {B} (x;,a,b)}{mathrm {B} (a,b)}}.}

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≤k)=I1−p(n−k,k+1)=1−Ip(k+1,n−k).{displaystyle F(k;,n,p)=Pr left(Xleq kright)=I_{1-p}(n-k,k+1)=1-I_{p}(k+1,n-k).}{displaystyle F(k;,n,p)=Pr left(Xleq kright)=I_{1-p}(n-k,k+1)=1-I_{p}(k+1,n-k).}


Properties


I0(a,b)=0I1(a,b)=1Ix(a,1)=xaIx(1,b)=1−(1−x)bIx(a,b)=1−I1−x(b,a)Ix(a+1,b)=Ix(a,b)−xa(1−x)baB(a,b)Ix(a,b+1)=Ix(a,b)+xa(1−x)bbB(a,b)B(x;a,b)=(−1)aB(xx−1;a,1−a−b){displaystyle {begin{aligned}I_{0}(a,b)&=0\I_{1}(a,b)&=1\I_{x}(a,1)&=x^{a}\I_{x}(1,b)&=1-(1-x)^{b}\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}}{amathrm {B} (a,b)}}\I_{x}(a,b+1)&=I_{x}(a,b)+{frac {x^{a}(1-x)^{b}}{bmathrm {B} (a,b)}}\mathrm {B} (x;a,b)&=(-1)^{a}mathrm {B} left({frac {x}{x-1}};a,1-a-bright)end{aligned}}}{displaystyle {begin{aligned}I_{0}(a,b)&=0\I_{1}(a,b)&=1\I_{x}(a,1)&=x^{a}\I_{x}(1,b)&=1-(1-x)^{b}\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}}{amathrm {B} (a,b)}}\I_{x}(a,b+1)&=I_{x}(a,b)+{frac {x^{a}(1-x)^{b}}{bmathrm {B} (a,b)}}\mathrm {B} (x;a,b)&=(-1)^{a}mathrm {B} left({frac {x}{x-1}};a,1-a-bright)end{aligned}}}


Multivariate beta function


The beta function can be extended to a function with more than two arguments:


B(α1,α2,…αn)=Γ(α1)Γ(α2)⋯Γ(αn)Γ(α1+α2+⋯+αn).{displaystyle mathrm {B} (alpha _{1},alpha _{2},ldots alpha _{n})={frac {Gamma (alpha _{1}),Gamma (alpha _{2})cdots Gamma (alpha _{n})}{Gamma (alpha _{1}+alpha _{2}+cdots +alpha _{n})}}.}{displaystyle mathrm {B} (alpha _{1},alpha _{2},ldots alpha _{n})={frac {Gamma (alpha _{1}),Gamma (alpha _{2})cdots Gamma (alpha _{n})}{Gamma (alpha _{1}+alpha _{2}+cdots +alpha _{n})}}.}

This multivariate beta function is used in the definition of the Dirichlet distribution.



Software implementation


Even if unavailable directly, the complete and incomplete beta function values can be calculated using functions commonly included in spreadsheet or computer algebra systems. In Excel, for example, the complete beta value can be calculated from the GammaLn function:


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

An incomplete beta value can be calculated as:



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

These result follow from the properties listed above.


Similarly, betainc (incomplete beta function) in MATLAB and GNU Octave, pbeta (probability of beta distribution) in R, or special.betainc in Python's SciPy package 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. In Mathematica, Beta[x, a, b] and BetaRegularized[x, a, b] give B(x;a,b){displaystyle mathrm {B} (x;,a,b)}{displaystyle mathrm {B} (x;,a,b)} and Ix(a,b){displaystyle I_{x}(a,b)}{displaystyle I_{x}(a,b)}, respectively.



See also



  • Beta distribution

  • Binomial distribution

  • Beta-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





  1. ^ ab Davis (1972) 6.2.2 p.258


  2. ^ Davis (1972) 6.2.1 p.258


  3. ^ Artin, Emil. The Gamma Function (PDF). pp. 18–19..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}





  • Askey, R. A.; Roy, R. (2010), "Beta function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248


  • Zelen, M.; Severo, N. C. (1972), "26. Probability functions", in Abramowitz, Milton; Stegun, Irene A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover Publications, pp. 925–995, ISBN 978-0-486-61272-0


  • Davis, Philip J. (1972), "6. Gamma function and related functions", in Abramowitz, Milton; Stegun, Irene A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover Publications, ISBN 978-0-486-61272-0


  • Paris, R. B. (2010), "Incomplete beta functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248


  • Press, W. H.; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 6.1 Gamma Function, Beta Function, Factorials", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8



External links




  • Hazewinkel, Michiel, ed. (2001) [1994], "Beta-function", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4


  • "Evaluation of beta function using Laplace transform". PlanetMath.

  • 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











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Wiciokrzew

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Wiciokrzew, suchodrzew Morfologia (wiciokrzew pospolity) Systematyka [1] Domena eukarionty Królestwo rośliny Klad rośliny naczyniowe Klad rośliny nasienne Klasa okrytonasienne Klad astrowe Rząd szczeciowce Rodzina przewiertniowate Rodzaj wiciokrzew Nazwa systematyczna Lonicera L. Sp. Pl. 173. 1 Mai 1753 Typ nomenklatoryczny Lonicera caprifolium L. [2] Multimedia w Wikimedia Commons Hasło w Wikisłowniku Kwiaty suchodrzewu pospolitego Owoce suchodrzewu pospolitego Kwiaty wiciokrzewu pomorskiego Kwiaty wiciokrzewu przewiercienia Wiciokrzew , suchodrzew ( Lonicera L.) – rodzaj roślin wieloletnich należący do rodziny przewiertniowatych ( Caprifoliaceae ). Rośliny zielne i pnące nazywane są wiciokrzewami, natomiast krzewy i niewielkie drzewa – suchodrzewami. Rodzaj liczy około 180 gatunków szeroko rozprzestrzenionych na całej półkuli północnej [3] [4]
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