tests/rtest_integrate_special.mac

   1 /******************************************************************************
   2   rtest_integrate_special.mac
   3   Integrals with special functions
   4 ******************************************************************************/
   5
   6 kill(all);
   7 done;
   8
   9 (reset(domain), reset(radexpand), done);
  10 done;
  11
  12 /* Integrals for the Incomplete Gamma function */
  13
  14 integrate(gamma_incomplete(a,z),z);
  15 z*gamma_incomplete(a,z)-gamma_incomplete(a+1,z);
  16
  17 integrate(z^(v-1)*gamma_incomplete(a,z),z);
  18 'integrate(z^(v-1)*gamma_incomplete(a,z),z);
  19
  20 /* Correct noun form if first argument is an expression.
  21  * See bug report Bug ID: 2968174 - Integration of hypergeometric bug */
  22 integrate(gamma_incomplete(2*a,z^2)*z^2,z);
  23 'integrate(gamma_incomplete(2*a,z^2)*z^2,z);
  24
  25 /* Integrals for Exponential Integral E(v,z) */
  26
  27 integrate(expintegral_e(v,a*z),z);
  28 -expintegral_e(v+1,a*z)/a;
  29
  30 integrate(expintegral_e(v,z),z);
  31 -expintegral_e(v+1,z);
  32
  33 integrate(z^(p-1)*expintegral_e(v,a*z),z);
  34 'integrate(z^(p-1)*expintegral_e(v,a*z),z);
  35
  36 /* Integrals for Exponential Integral E1(z) */
  37
  38 integrate(expintegral_e1(a*z),z);
  39 -expintegral_e(2,a*z)/a;
  40
  41 integrate(expintegral_e1(z),z);
  42 -expintegral_e(2,z);
  43
  44 integrate(z^(p-1)*expintegral_e1(a*z),z);
  45 'integrate(z^(p-1)*expintegral_e1(a*z),z);
  46
  47 /* Integrals for Exponential Integral Ei(z) */
  48
  49 integrate(expintegral_ei(z),z);
  50 z*expintegral_ei(z)-%e^z;
  51
  52 integrate(expintegral_ei(a*z),z);
  53 (a*z*expintegral_ei(a*z)-%e^(a*z))/a;
  54
  55 integrate(expintegral_ei(a*z+b),z);
  56 ((a*z+b)*expintegral_ei(a*z+b)-%e^(a*z+b))/a;
  57
  58 integrate(z^(p-1)*expintegral_ei(a*z),z);
  59 'integrate(z^(p-1)*expintegral_ei(a*z),z);
  60
  61 integrate(z*expintegral_ei(a*z),z);
  62 z^2*expintegral_ei(a*z)/2-(a*z-1)*%e^(a*z)/(2*a^2);
  63
  64 integrate(expintegral_ei(a*z)/z,z);
  65 'integrate(expintegral_ei(a*z)/z,z);
  66
  67 integrate(expintegral_ei(a*z)/z^2,z);
  68 gamma_incomplete(-1,-a*z)*a-expintegral_ei(a*z)/z;
  69
  70 integrate(expintegral_ei(a*z+b)/z^2,z);
  71 'integrate(expintegral_ei(a*z+b)/z^2,z);
  72
  73 integrate(z^(p-1)*expintegral_ei(a*z^r),z);
  74 'integrate(z^(p-1)*expintegral_ei(a*z^r),z);
  75
  76 integrate(z^n*%e^(b*z)*expintegral_ei(a*z),z);
  77 'integrate(z^n*%e^(b*z)*expintegral_ei(a*z),z);
  78
  79 integrate(z*%e^(b*z)*expintegral_ei(a*z),z);
  80 'integrate(z*%e^(b*z)*expintegral_ei(a*z),z);
  81
  82 integrate(z^2*%e^(b*z)*expintegral_ei(a*z),z);
  83 'integrate(z^2*%e^(b*z)*expintegral_ei(a*z),z);
  84
  85 integrate(z^3*%e^(b*z)*expintegral_ei(a*z),z);
  86 integrate(z^3*%e^(b*z)*expintegral_ei(a*z),z);
  87
  88 integrate(%e^(a*z)*expintegral_ei(a*z)/z,z);
  89 1/2*expintegral_ei(a*z)^2;
  90
  91 /* More integrals after extending the integrator
  92    do look for argument substitution of special functions */
  93
  94 integrate(expintegral_ei(x^2),x);
  95 x*expintegral_ei(x^2)+sqrt(%pi)*%i*erf(%i*x);
  96
  97 integrate(expintegral_ei(x^3),x);
  98 x*expintegral_ei(x^3)-gamma_incomplete(1/3,-x^3);
  99
 100 integrate(expintegral_ei(x^-2),x);
 101 %i*gamma_incomplete(-1/2,-1/x^2)*x/abs(x)+expintegral_ei(1/x^2)*x;
 102
 103 integrate(expintegral_ei(x^(1/2)),x);
 104 2*(expintegral_ei(sqrt(x))*x/2-(sqrt(x)-1)*%e^sqrt(x)/2);
 105
 106 integrate(expintegral_ei(x^(-1/2)),x);
 107 2*(expintegral_ei(1/sqrt(x))*x/2+gamma_incomplete(-2,-1/sqrt(x))/2);
 108
 109 integrate(expintegral_ei(x^(-3/4)),x);
 110 4*(expintegral_ei(1/x^(3/4))*x/4+gamma_incomplete(-4/3,-1/x^(3/4))/4);
 111
 112 integrate(expintegral_ei((x+1)^(1/2)),x);
 113 2*((x+1)*expintegral_ei(sqrt(x+1))/2-(sqrt(x+1)-1)*%e^sqrt(x+1)/2);
 114
 115 /* Integrals for Exponential Integral Li(z) */
 116
 117 integrate(expintegral_li(a*z),z);
 118 (a*z*expintegral_li(a*z)-expintegral_ei(2*log(a*z)))/a;
 119
 120 integrate(expintegral_li(z),z);
 121 z*expintegral_li(z)-expintegral_ei(2*log(z));
 122
 123 integrate(z*expintegral_li(z),z);
 124 z^2*expintegral_li(z)/2+gamma_incomplete(0,-3*log(z))/2;
 125
 126 integrate(sqrt(z)*expintegral_li(z),z),logexpand;
 127 2*(z^(3/2)*expintegral_li(z)/3+gamma_incomplete(0,-5/2*log(z))/3);
 128
 129 /* Integrals for Exponential Integral Si(z) */
 130
 131 integrate(expintegral_si(z),z);
 132 z*expintegral_si(z)+cos(z);
 133
 134 integrate(z*expintegral_si(z),z);
 135 z^2*expintegral_si(z)/2 - (sin(z)-z*cos(z)) /2;
 136
 137 /* Integrals for Exponential Integral Ci(z) */
 138
 139 integrate(expintegral_ci(z),z);
 140 z*expintegral_ci(z)-sin(z);
 141
 142 integrate(z*expintegral_ci(z),z);
 143 z^2*expintegral_ci(z)/2 - (z*sin(z)+cos(z)) /2;
 144
 145 /* Integrals for Exponential Integral Shi(z) */
 146
 147 integrate(expintegral_shi(z),z);
 148 z*expintegral_shi(z)-cosh(z);
 149
 150 integrate(z*expintegral_shi(z),z);
 151 z^2*expintegral_shi(z)/2 - ((z-1)*%e^z-(-z-1)*%e^(-z))/4;
 152
 153 /* Integrals for Exponential Integral Chi(z) */
 154
 155 integrate(expintegral_chi(z),z);
 156 z*expintegral_chi(z)-sinh(z);
 157
 158 integrate(z*expintegral_chi(z),z);
 159 z^2*expintegral_chi(z)/2 - ((z-1)*%e^z+(-z-1)*%e^(-z))/4;
 160
 161 /* Integrals of Lambert W function.  This checks that other parts
 162    of the integration routines work with the 'integral property */
 163
 164 /* This is tested in rtest_lambert_w.mac */
 165 /* integrate(lambert_w(x),x);
 166 x*(lambert_w(x)^2-lambert_w(x)+1)/lambert_w(x); */
 167
 168 integrate(lambert_w(a*x+b),x);
 169 (a*x+b)*(lambert_w(a*x+b)^2-lambert_w(a*x+b)+1)/(a*lambert_w(a*x+b));
 170
 171 integrate(sin(x)*lambert_w(cos(x)),x);
 172 -(cos(x)*(lambert_w(cos(x))^2-lambert_w(cos(x))+1)/lambert_w(cos(x)));
 173
 174 integrate(diff(f(x),x)*lambert_w(f(x)),x);
 175  f(x)*(lambert_w(f(x))^2-lambert_w(f(x))+1)/lambert_w(f(x));
 176
 177 /* Integrals where the property is a function */
 178 integrate(bessel_j(0,x),x);
 179 x*(bessel_j(0,x)*(2-%pi*struve_h(1,x))+%pi*bessel_j(1,x)*struve_h(0,x))/2;
 180
 181 integrate(bessel_j(1,x),x);
 182 -bessel_j(0,x);
 183
 184 integrate(bessel_j(2,x),x);
 185 x^3/24*hypergeometric([3/2],[5/2,3],-x^2/4);
 186
 187 integrate(bessel_j(n,x),n);
 188 'integrate(bessel_j(n,x),n);
 189
 190 /* Examples of a subscripted function */
 191 integrate(psi[0](x),x);
 192 log_gamma(x);
 193
 194 integrate(psi[1](x),x);
 195 psi[0](x);