卷青白韭菜队-睡不够day16-matlab第十一章使用特殊函数

[睡不够123]

贝塔函数：x = beta(m,n)

贝塔函数可以用来计算含有sinθ和cosθ乘积的积分。

∫sin^(2m-1) θ*cos^(2n-1) θdθ = 1/2B(m,n) 因此：∫sin^5 θcos^3 θdθ = 1/2B(3,2) ==> 0.5*beta(3,2)

2m-1=5,2n-1=3; m=3,n=2

>> 0.5 * beta(3,2)
ans =
    0.0417
>> 

其他三角函数积分可以转化为sin和cos的形式来求

特殊积分：

幂积分：y = expint(x)   当expint(0) = inf

其他特殊形式的积分可以通过help mfunlist命令查询

>> help mfunlist
 mfunlist Special functions for MFUN.
    mfunlist will be removed in a future release. 
    Instead, use the appropriate special function listed below.
    For example, use bernoulli(n) instead of mfun('bernoulli',n).
 
    The following special functions are listed in alphabetical order
    according to the third column. n denotes an integer argument, 
    x denotes a real argument, and z denotes a complex argument. For 
    more detailed descriptions of the functions, including any argument
    restrictions, see the documentation of the active symbolic engine.
 
 bernoulli      n       Bernoulli Numbers                                     => bernoulli(n)
 bernoulli      n,z     Bernoulli Polynomials                                 => bernoulli(n,z)
 BesselI        x1,x    Bessel Function of the First Kind                     => besseli(v,x)
 BesselJ        x1,x    Bessel Function of the First Kind                     => besselj(v,x)
 BesselK        x1,x    Bessel Function of the Second Kind                    => besselk(v,x)
 BesselY        x1,x    Bessel Function of the Second Kind                    => bessely(v,x)
 Beta           z1,z2   Beta Function                                         => beta(x,y)
 binomial       x1,x2   Binomial Coefficients                                 => nchoosek(m,n)
 EllipticF -    z,k     Incomplete Elliptic Integral, First Kind              => ellipticF(z,k)
 EllipticK -    k       Complete Elliptic Integral, First Kind                => ellipticK(k)
 EllipticCK -   k       Complementary Complete Integral, First Kind           => ellipticCK(k)
 EllipticE -    k       Complete Elliptic Integrals, Second Kind              => ellipticE(k)
 EllipticE -    z,k     Incomplete Elliptic Integrals, Second Kind            => ellipticE(z,k)
 EllipticCE -   k       Complementary Complete Elliptic Integral, Second Kind => ellipticCE(k)
 EllipticPi -   nu,k    Complete Elliptic Integrals, Third Kind               => ellipticPi(nu,k)
 EllipticPi -   z,nu,k  Incomplete Elliptic Integrals, Third Kind             => ellipticPi(z,nu,k)
 EllipticCPi -  nu,k    Complementary Complete Elliptic Integral, Third Kind  => ellipticCPi(nu,k)
 erfc           z       Complementary Error Function                          => erfc(z)
 erfc           n,z     Complementary Error Function's Iterated Integrals     => erfc(n,z)
 Ci             z       Cosine Integral                                       => sinint(z)
 dawson         x       Dawson's Integral                                     => dawson(z)
 Psi            z       Digamma Function                                      => psi(z)
 dilog          x       Dilogarithm Integral                                  => dilog(x)
 erf            z       Error Function                                        => erf(z)
 euler          n       Euler Numbers                                         => euler(n)
 euler          n,z     Euler Polynomials                                     => euler(n,z)
 Ei             x       Exponential Integral                                  => ei(n)
 Ei             n,z     Exponential Integral                                  => expint(n,z)
 FresnelC       x       Fresnel Cosine Integral                               => fresnelc(x)
 FresnelS       x       Fresnel Sine Integral                                 => fresnels(x)
 GAMMA          z       Gamma Function                                        => gamma(z)
 harmonic       n       Harmonic Function                                     => harmonic(n)
 Chi            z       Hyperbolic Cosine Integral                            => coshint(z)
 Shi            z       Hyperbolic Sine Integral                              => sinhint(z)
 GAMMA          z1,z2   Incomplete Gamma Function                             => igamma(z1,z2)
 L              n,x     Laguerre                                              => laguerreL(n,x)
 L              n,x1,x  Generalized Laguerre                                  => laguerreL(n,x1,x)
 W              z       Lambert's W Function                                  => lambertw(z)
 W              n,z     Lambert's W Function                                  => lambertw(n,z)
 lnGAMMA        z       Logarithm of the Gamma function                       => gammaln(z)
 Li             x       Logarithmic Integral                                  => logint(x)
 Psi            n,z     Polygamma Function                                    => psi(n,z)
 Ssi            z       Shifted Sine Integral                                 => ssinint(z)
 Si             z       Sine Integral                                         => sinint(z)
 Zeta           z       (Riemann) Zeta Function                               => zeta(z)
 Zeta           n,z     (Riemann) Zeta Function                               => zeta(n,z)
 
        Orthogonal Polynomials
 T      n,x             Chebyshev of the First Kind                           => chebyshevT(n,x)
 U      n,x             Chebyshev of the Second Kind                          => chebyshevU(n,x)
 G      n,x1,x          Gegenbauer                                            => gegenbauerC(n,x1,x)
 H      n,x             Hermite                                               => hermiteH(n,x)
 P      n,x1,x2,x       Jacobi                                                => jacobiP(n,x1,x2,x)
 P      n,x             Legendre                                              => legendreP(n,x)
 
    See also mfun, symengine.

    mfunlist 的参考页

勒让德函数：p = legendre(n,x)，相伴勒让德方程只在(-1，1)之间有效

>> x = linspace(-1,1);
>> p1 = legendre(1,x);p2=legendre(2,x);p3 = legendre(3,x);p4 = legendre(4,x);
>> plot(x,p1),xlabel('x'),ylabel('p1')

勒让德函数n =2 时图像 

 勒让德函数相伴的三个函数图像

>> f = p2(1,:);
>> g = p2(2,:);
>> h = p2(3,:);
>> plot(x,f,x,g,'--',x,h,'b:')
>> 

 亚里函数：

>> x= linspace(-10,5);
>> y = airy(x);
>> plot(x,y),grid on