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erf-exp-expand-inf-integrate-load-romberg-tex

f(x):=exp(-x^2);

tex( expand( integrat...

tex(integrate( expand...

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erf-plot3d-product-sqrt
plot3d( Z(10,x), [n, 1, 50], [x, 0, 100]);

G(x):=0.5*(1+erf(x/sq...

Q(x,xm,xd):= G((x-xm)...

S(k,x,xm,xd):= Q(x,k*...

Calculate

erf-exp-pi-realroots-sqrt

gaus(x) := 1/(q*sqrt(...

efs(x) := 1/2*(1+erf(...

realroots(efs(-5*60)=...

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erf-exp-pi-realroots-sqrt

gaus(x) := 1/(q*sqrt(...

efs(x) := 1/2*(1+erf(...

realroots(efs(-5*60)=...

Calculate

erf-exp-float-pi-realroots-sqrt

gaus(x) := float(1/(q...

efs(x) := float(1/2*(...

realroots(efs(-5*60)=...

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erf-float-sqrt

a:2;

float(erf(1/sqrt(20)));

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erf-plot3d-sqrt
plot3d([ [(0.5*(1+erf((x-4)/(50*sqrt(2)))))^n,[x,-100,100],[n,1,500]]]);

plot3d([ [(0.5*(1+erf...

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erf-exp-plot2d
plot2d([(erf(3.44688-1.26491*(t-1))-erf(3.44688))*exp((1.6*(t-1)-8.72)*(t-1))*3.23957E+7],[t,1,2.0]);

plot2d([(erf(3.44688-...

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erf-exp-sqrt

rombergit:1000;

romberg(y*exp(-(y-1)^...

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erf

Run Example
(%i1)f(x):=exp(-x^2);
                                              2
(%o1)                          f(x) := exp(- x )
(%i2) tex( expand( integrate( (x-y-l)*f(x), x, l+y, inf) ));
$${{\sqrt{\pi}\,y\,\mathrm{erf}\left(y+l\right)}\over{2}}+{{\sqrt{\pi
 }\,l\,\mathrm{erf}\left(y+l\right)}\over{2}}+{{e^{-y^2-2\,l\,y-l^2}
 }\over{2}}-{{\sqrt{\pi}\,y}\over{2}}-{{\sqrt{\pi}\,l}\over{2}}$$
(%o2)                                false
(%i3) tex(integrate( expand( integrate( (x-y-l)*f(x), x, l+y, inf) * f(y)),y, -inf,inf));
$$\int_{-\infty }^{\infty }{{{\sqrt{\pi}\,y\,e^ {- y^2 }\,
 \mathrm{erf}\left(y+l\right)}\over{2}}+{{\sqrt{\pi}\,l\,e^ {- y^2 }
 \,\mathrm{erf}\left(y+l\right)}\over{2}}-{{\sqrt{\pi}\,y\,e^ {- y^2
  }}\over{2}}-{{\sqrt{\pi}\,l\,e^ {- y^2 }}\over{2}}+{{e^{-2\,y^2-2\,
 l\,y-l^2}}\over{2}}\;dy}$$
(%o3)                                false
(%i4) integrate( expand( integrate( (x-y-l)*f(x), x, l+y, inf) * f(y)),y, -inf,inf);
       inf                     2                               2
      /                     - y                             - y
      [       sqrt(%pi) y %e     erf(y + l)   sqrt(%pi) l %e     erf(y + l)
(%o4) I      (----------------------------- + -----------------------------
      ]                     2                               2
      /
       - inf
                             2                    2          2            2
                          - y                  - y      - 2 y  - 2 l y - l
            sqrt(%pi) y %e       sqrt(%pi) l %e       %e
          - ------------------ - ------------------ + ---------------------) dy
                    2                    2                      2
(%i5) integrate( y*exp(-y^2), y, -inf,inf);
(%o5)                                  0
(%i6) integrate( exp(-y^2)*erf(y+l),y,-inf,inf );
                           inf
                          /           2
                          [        - y
(%o6)                     I      %e     erf(y + l) dy
                          ]
                          /
                           - inf
(%i7) integrate( y*exp(-y^2),y,-inf,inf);
(%o7)                                  0
(%i8) integrate( exp(-y^2),y,-inf,inf);
(%o8)                              sqrt(%pi)
(%i9) integrate( exp(-(y+l)^2),y,-inf,inf);
(%o9)                              sqrt(%pi)
(%i10) romberg(exp(-y^2)*erf(y+l),y,-inf,inf);
                              2
                           - y
(%o10)           romberg(%e     erf(y + l), y, - 1.0 inf, inf)
(%i11) 
Run Example
G(x):=0.5*(1+erf(x/sqrt(2)));
                                                x
(%o1)                   G(x) := 0.5 (1 + erf(-------))
                                             sqrt(2)
(%i2) Q(x,m,d):= G((x-m)/d);
                                            x - m
(%o2)                       Q(x, m, d) := G(-----)
                                              d
(%i3) 
Run Example
eq1:exp(-x^2);
                                         2
                                      - x
(%o1)                               %e
(%i2) integrate(eq1,x);
                               sqrt(%pi) erf(x)
(%o2)                          ----------------
                                      2
(%i3) erf(2);
(%o3)                               erf(2)
(%i4) 

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