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letsimp-minf

letsimp(1/x, minf);

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letsimp

letsimp(n!/n);

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letsimp

r=1/2 + 3^(1/4)/2^(1/...

letsimp(r^2);

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letsimp-minf

letsimp(1/x, minf);

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letsimp

letsimp(n!/n);

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letsimp

r=1/2 + 3^(1/4)/2^(1/...

letsimp(r^2);

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letsimp

Run Example
(%i1)matchdeclare(a, true);
(%o1)                                done
(%i2) let(cot(a), cos(a)/sin(a));
                                          cos(a)
(%o2)                          cot(a) --> ------
                                          sin(a)
(%i3) letsimp(cot(z));
                                    cos(z)
(%o3)                               ------
                                    sin(z)
(%i4) letsimp(tan(z));
(%o4)                               tan(z)
(%i5) 
Run Example
/* Standard (x,y)->
 (r,theta) coordinate translation */x(t) := r(t) * cos(theta(t));
(%o1)                     x(t) := r(t) cos(theta(t))
(%i2) y(t) := r(t) * sin(theta(t));
(%o2)                     y(t) := r(t) sin(theta(t))
(%i3) /* Constant course assumption */declare (slope,constant);
(%o3)                                done
(%i4) declare (slope,real);
(%o4)                                done
(%i5) declare (yintercept,constant);
(%o5)                                done
(%i6) declare (yintercept, real);
(%o6)                                done
(%i7) eq1: y(t)=slope * x(t) + yintercept;
(%o7)     r(t) sin(theta(t)) = slope r(t) cos(theta(t)) + yintercept
(%i8) /* Known & constant sound velocity */declare (soundvel,constant);
(%o8)                                done
(%i9) assume(soundvel>
0);
(%o9)                           [0 < soundvel]
(%i10) /* Known and constant base frequency */declare (basefrequency, constant);
(%o10)                               done
(%i11) assume(basefrequency>
0);
(%o11)                        [0 < basefrequency]
(%i12) /* f-- observed frequency. Standard doppler equation*/declare(f, real);
(%o12)                               done
(%i13) let (diff(r(t),t), (f(t)/basefrequency-1)*soundvel);
                  d                           f(t)
(%o13)            -- (r(t)) --> soundvel (------------- - 1)
                  dt                      basefrequency
(%i14) /* Differentiate eq1 and solve for r(t) */eq3: letsimp(solve([diff(y(t),t) = slope * diff(x(t),t)],r(t)));
(%o14) [r(t) = 
                     slope soundvel cos(theta(t))
- -------------------------------------------------------------------
                       d                               d
  slope sin(theta(t)) (-- (theta(t))) + cos(theta(t)) (-- (theta(t)))
                       dt                              dt
                         soundvel sin(theta(t))
 + -------------------------------------------------------------------
                        d                               d
   slope sin(theta(t)) (-- (theta(t))) + cos(theta(t)) (-- (theta(t)))
                        dt                              dt
 + (slope soundvel f(t) cos(theta(t)))/(basefrequency slope sin(theta(t))
  d                                             d
 (-- (theta(t))) + basefrequency cos(theta(t)) (-- (theta(t))))
  dt                                            dt
 + (- (soundvel f(t) sin(theta(t)))/(basefrequency slope sin(theta(t))
  d                                             d
 (-- (theta(t))) + basefrequency cos(theta(t)) (-- (theta(t)))))]
  dt                                            dt
(%i15) /* Differentiate eq3 and solve for slope */eq6: trigsimp(solve(letsimp(diff(eq3,t)), slope));
                        2                                          2
(%o15) [slope = (sqrt((f (t) - 2 basefrequency f(t) + basefrequency )
   2
  d              2                               d           d
 (--- (theta(t)))  + (2 basefrequency - 2 f(t)) (-- (f(t))) (-- (theta(t)))
    2                                            dt          dt
  dt
   2
  d                       2                                             2
 (--- (theta(t))) + (- 8 f (t) + 16 basefrequency f(t) - 8 basefrequency )
    2
  dt
  d             4    d         2  d             2
 (-- (theta(t)))  + (-- (f(t)))  (-- (theta(t))) )
  dt                 dt           dt
                                  2
 + ((2 f(t) - 2 basefrequency) sin (theta(t)) - f(t) + basefrequency)
   2
  d
 (--- (theta(t))) + (2 basefrequency - 2 f(t)) cos(theta(t)) sin(theta(t))
    2
  dt
  d             2    d              d             2             d
 (-- (theta(t)))  + (-- (f(t)) - 2 (-- (f(t))) sin (theta(t))) (-- (theta(t))))
  dt                 dt             dt                          dt
                                                           2
                                                          d
/((2 f(t) - 2 basefrequency) cos(theta(t)) sin(theta(t)) (--- (theta(t)))
                                                            2
                                                          dt
                                  2
 + ((2 f(t) - 2 basefrequency) sin (theta(t)) + 2 f(t) - 2 basefrequency)
  d             2      d
 (-- (theta(t)))  - 2 (-- (f(t))) cos(theta(t)) sin(theta(t))
  dt                   dt
  d                                  2
 (-- (theta(t)))), slope = - (sqrt((f (t) - 2 basefrequency f(t)
  dt
                     2
                2   d              2                               d
 + basefrequency ) (--- (theta(t)))  + (2 basefrequency - 2 f(t)) (-- (f(t)))
                      2                                            dt
                    dt
                   2
  d               d                       2
 (-- (theta(t))) (--- (theta(t))) + (- 8 f (t) + 16 basefrequency f(t)
  dt                2
                  dt
                  2   d             4    d         2  d             2
 - 8 basefrequency ) (-- (theta(t)))  + (-- (f(t)))  (-- (theta(t))) )
                      dt                 dt           dt
                                  2
 + ((2 f(t) - 2 basefrequency) cos (theta(t)) - f(t) + basefrequency)
   2
  d
 (--- (theta(t))) + (2 f(t) - 2 basefrequency) cos(theta(t)) sin(theta(t))
    2
  dt
  d             2    d              d             2             d
 (-- (theta(t)))  + (-- (f(t)) - 2 (-- (f(t))) cos (theta(t))) (-- (theta(t))))
  dt                 dt             dt                          dt
                                                           2
                                                          d
/((2 f(t) - 2 basefrequency) cos(theta(t)) sin(theta(t)) (--- (theta(t)))
                                                            2
                                                          dt
                                  2
 + ((2 basefrequency - 2 f(t)) cos (theta(t)) + 4 f(t) - 4 basefrequency)
  d             2      d                                       d
 (-- (theta(t)))  - 2 (-- (f(t))) cos(theta(t)) sin(theta(t)) (-- (theta(t))))]
  dt                   dt                                      dt
(%i16) /* Now we have a number of estimates for the slope constant. *//* We can use any statistical technique to estimate the actual constant *//* Constant speed assumption */eq7: letsimp(diff(letsimp(diff(x(t),t)),t)) = 0;
                                             2
                                            d
(%o16) (- basefrequency r(t) sin(theta(t)) (--- (theta(t)))
                                              2
                                            dt
                                     d             2
 - basefrequency r(t) cos(theta(t)) (-- (theta(t)))
                                     dt
                                  d
 - 2 soundvel f(t) sin(theta(t)) (-- (theta(t)))
                                  dt
                                           d
 + 2 basefrequency soundvel sin(theta(t)) (-- (theta(t)))
                                           dt
             d
 + soundvel (-- (f(t))) cos(theta(t)))/basefrequency = 0
             dt
(%i17) eq8: solve(eq7, r(t));
(%o17) [r(t) = - ((2 soundvel f(t) - 2 basefrequency soundvel) sin(theta(t))
  d                          d
 (-- (theta(t))) - soundvel (-- (f(t))) cos(theta(t)))
  dt                         dt
                                2
                               d
/(basefrequency sin(theta(t)) (--- (theta(t)))
                                 2
                               dt
                                d             2
 + basefrequency cos(theta(t)) (-- (theta(t))) )]
                                dt
(%i18) eq9: subst(eq8, eq1);
(%o18) - (sin(theta(t)) ((2 soundvel f(t) - 2 basefrequency soundvel)
                d                          d
 sin(theta(t)) (-- (theta(t))) - soundvel (-- (f(t))) cos(theta(t))))
                dt                         dt
                                2
                               d
/(basefrequency sin(theta(t)) (--- (theta(t)))
                                 2
                               dt
                                d             2
 + basefrequency cos(theta(t)) (-- (theta(t))) ) = 
                                dt
yintercept - (slope cos(theta(t)) ((2 soundvel f(t) - 2 basefrequency soundvel)
                d                          d
 sin(theta(t)) (-- (theta(t))) - soundvel (-- (f(t))) cos(theta(t))))
                dt                         dt
                                2
                               d
/(basefrequency sin(theta(t)) (--- (theta(t)))
                                 2
                               dt
                                d             2
 + basefrequency cos(theta(t)) (-- (theta(t))) )
                                dt
(%i19) eq10: solve(eq9, yintercept);
(%o19) [yintercept = - (((2 soundvel f(t) - 2 basefrequency soundvel)
    2
 sin (theta(t)) + (2 basefrequency slope soundvel - 2 slope soundvel f(t))
                               d
 cos(theta(t)) sin(theta(t))) (-- (theta(t)))
                               dt
             d
 - soundvel (-- (f(t))) cos(theta(t)) sin(theta(t))
             dt
                   d             2
 + slope soundvel (-- (f(t))) cos (theta(t)))
                   dt
                                2
                               d
/(basefrequency sin(theta(t)) (--- (theta(t)))
                                 2
                               dt
                                d             2
 + basefrequency cos(theta(t)) (-- (theta(t))) )]
                                dt
(%i20) /* Now we have a number of estimates for the yintercept constant. *//* We can use any statistical technique to estimate the actual constant *//* This gives us r(theta) in terms of slope and y_intercept*/solve(eq1, r(t));
                                     yintercept
(%o20)           [r(t) = -----------------------------------]
                         sin(theta(t)) - slope cos(theta(t))
(%i21) 
Run Example
x(t) := r(t) * sin(theta(t));
(%o1)                     x(t) := r(t) sin(theta(t))
(%i2) y(t) := r(t) * cos(theta(t));
(%o2)                     y(t) := r(t) cos(theta(t))
(%i3) declare (slope,constant);
(%o3)                                done
(%i4) declare (xvel, constant);
(%o4)                                done
(%i5) declare (slope,real);
(%o5)                                done
(%i6) declare (xvel, real);
(%o6)                                done
(%i7) diff(x(t),t) = xvel;
                          d                 d
(%o7) r(t) cos(theta(t)) (-- (theta(t))) + (-- (r(t))) sin(theta(t)) = xvel
                          dt                dt
(%i8) diff(y(t),t) = xvel * slope;
       d                                              d
(%o8) (-- (r(t))) cos(theta(t)) - r(t) sin(theta(t)) (-- (theta(t))) = 
       dt                                             dt
                                                                     slope xvel
(%i9) declare (soundvel,constant);
(%o9)                                done
(%i10) declare (soundvel,posfun);
(%o10)                               done
(%i11) declare (f0, constant);
(%o11)                               done
(%i12) eq20: diff(diff(y(t),t),t) = 0;
                              2
                             d
(%o12) - r(t) sin(theta(t)) (--- (theta(t)))
                               2
                             dt
                       d             2      d
 - r(t) cos(theta(t)) (-- (theta(t)))  - 2 (-- (r(t))) sin(theta(t))
                       dt                   dt
                     2
  d                 d
 (-- (theta(t))) + (--- (r(t))) cos(theta(t)) = 0
  dt                  2
                    dt
(%i13) let (diff(r(t),t), (f(t)/f0-1)*soundvel);
                       d                       f(t)
(%o13)                 -- (r(t)) --> soundvel (---- - 1)
                       dt                       f0
(%i14) eq30: diff(y(t),t) = slope * diff(x(t),t);
        d                                              d
(%o14) (-- (r(t))) cos(theta(t)) - r(t) sin(theta(t)) (-- (theta(t))) = 
        dt                                             dt
                                    d                 d
         slope (r(t) cos(theta(t)) (-- (theta(t))) + (-- (r(t))) sin(theta(t)))
                                    dt                dt
(%i15) eq40: letsimp(solve([eq30],r(t)));
(%o15) [r(t) = 
                        soundvel cos(theta(t))
- -------------------------------------------------------------------
                 d                                     d
  sin(theta(t)) (-- (theta(t))) + slope cos(theta(t)) (-- (theta(t)))
                 dt                                    dt
                      slope soundvel sin(theta(t))
 + -------------------------------------------------------------------
                  d                                     d
   sin(theta(t)) (-- (theta(t))) + slope cos(theta(t)) (-- (theta(t)))
                  dt                                    dt
                          soundvel f(t) cos(theta(t))
 + -------------------------------------------------------------------------
                     d                                        d
   f0 sin(theta(t)) (-- (theta(t))) + f0 slope cos(theta(t)) (-- (theta(t)))
                     dt                                       dt
 + (- (slope soundvel f(t) sin(theta(t)))
                    d                                        d
/(f0 sin(theta(t)) (-- (theta(t))) + f0 slope cos(theta(t)) (-- (theta(t)))))]
                    dt                                       dt
(%i16) eq50: letsimp(diff(eq40,t));
                     soundvel f(t)
(%o16) [- soundvel + ------------- = 
                          f0
                                         d
           slope soundvel cos(theta(t)) (-- (theta(t)))
                                         dt
-------------------------------------------------------------------
               d                                     d
sin(theta(t)) (-- (theta(t))) + slope cos(theta(t)) (-- (theta(t)))
               dt                                    dt
                                         d
                 soundvel sin(theta(t)) (-- (theta(t)))
                                         dt
 + -------------------------------------------------------------------
                  d                                     d
   sin(theta(t)) (-- (theta(t))) + slope cos(theta(t)) (-- (theta(t)))
                  dt                                    dt
                                d
                      soundvel (-- (f(t))) cos(theta(t))
                                dt
 + -------------------------------------------------------------------------
                     d                                        d
   f0 sin(theta(t)) (-- (theta(t))) + f0 slope cos(theta(t)) (-- (theta(t)))
                     dt                                       dt
                       d
 + (- (slope soundvel (-- (f(t))) sin(theta(t)))
                       dt
                    d                                        d
/(f0 sin(theta(t)) (-- (theta(t))) + f0 slope cos(theta(t)) (-- (theta(t)))))
                    dt                                       dt
                                          d
 + (- (slope soundvel f(t) cos(theta(t)) (-- (theta(t))))
                                          dt
                    d                                        d
/(f0 sin(theta(t)) (-- (theta(t))) + f0 slope cos(theta(t)) (-- (theta(t)))))
                    dt                                       dt
                                    d
 + (- (soundvel f(t) sin(theta(t)) (-- (theta(t))))
                                    dt
                    d                                        d
/(f0 sin(theta(t)) (-- (theta(t))) + f0 slope cos(theta(t)) (-- (theta(t)))))
                    dt                                       dt
                                            2
                                           d
 + (soundvel cos(theta(t)) (sin(theta(t)) (--- (theta(t)))
                                             2
                                           dt
                         2
                        d                                      d             2
 + slope cos(theta(t)) (--- (theta(t))) - slope sin(theta(t)) (-- (theta(t)))
                          2                                    dt
                        dt
                  d             2
 + cos(theta(t)) (-- (theta(t))) ))
                  dt
                 d                                     d              2
/(sin(theta(t)) (-- (theta(t))) + slope cos(theta(t)) (-- (theta(t))))
                 dt                                    dt
                                                     2
                                                    d
 + (- (slope soundvel sin(theta(t)) (sin(theta(t)) (--- (theta(t)))
                                                      2
                                                    dt
                         2
                        d                                      d             2
 + slope cos(theta(t)) (--- (theta(t))) - slope sin(theta(t)) (-- (theta(t)))
                          2                                    dt
                        dt
                  d             2
 + cos(theta(t)) (-- (theta(t))) ))
                  dt
                 d                                     d              2
/(sin(theta(t)) (-- (theta(t))) + slope cos(theta(t)) (-- (theta(t)))) )
                 dt                                    dt
                                                       2
                                                      d
 + (- (soundvel f(t) cos(theta(t)) (f0 sin(theta(t)) (--- (theta(t)))
                                                        2
                                                      dt
                            2
                           d
 + f0 slope cos(theta(t)) (--- (theta(t)))
                             2
                           dt
                           d             2
 - f0 slope sin(theta(t)) (-- (theta(t)))
                           dt
                     d             2
 + f0 cos(theta(t)) (-- (theta(t))) ))
                     dt
                    d                                        d              2
/(f0 sin(theta(t)) (-- (theta(t))) + f0 slope cos(theta(t)) (-- (theta(t)))) )
                    dt                                       dt
                                                          2
                                                         d
 + (slope soundvel f(t) sin(theta(t)) (f0 sin(theta(t)) (--- (theta(t)))
                                                           2
                                                         dt
                            2
                           d
 + f0 slope cos(theta(t)) (--- (theta(t)))
                             2
                           dt
                           d             2
 - f0 slope sin(theta(t)) (-- (theta(t)))
                           dt
                     d             2
 + f0 cos(theta(t)) (-- (theta(t))) ))
                     dt
                    d                                        d              2
/(f0 sin(theta(t)) (-- (theta(t))) + f0 slope cos(theta(t)) (-- (theta(t)))) ]
                    dt                                       dt
(%i17) 
[assume,constant,cos,declare,diff,expandwrt,globalsolve,let,letsimp,ratcoef,realonly,rhs,sin,subst,tlimswitch,trigsimp,true] [assume,constant,cos,declare,diff,expandwrt,let,letsimp,sin,solve,trigsimp] [assume,constant,cos,declare,diff,expandwrt_factored,globalsolve,let,letsimp,posfun,ratcoef,realonly,sin,subst,tlimswitch,trigsimp,true] [assume,constant,cos,declare,diff,factor,let,letsimp,sin,solve,trigsimp] [assume,constant,cos,declare,diff,factorout,globalsolve,let,letsimp,lhs,posfun,realonly,sin,subst,trigsimp,true] [assume,constant,cos,declare,diff,globalsolve,let,letsimp,lhs,posfun,ratsimp,realonly,sin,subst,trigsimp,true] [assume,constant,cos,declare,diff,globalsolve,let,letsimp,lhs,posfun,ratsubst,realonly,sin,subst,trigsimp,true] [assume,constant,cos,declare,diff,globalsolve,let,letsimp,lhs,posfun,realonly,sin,subst,trigsimp,true] [assume,constant,cos,declare,diff,globalsolve,let,letsimp,posfun,realonly,sin,subst,trigsimp,true] [assume,constant,cos,declare,diff,globalsolve,let,letsimp,posfun,realonly,sin,subst,true] [assume,constant,cos,declare,diff,let,letsimp,mainvar,sin,solve] [assume,constant,cos,declare,diff,let,letsimp,optimize,sin,solve,trigsimp] [assume,constant,cos,declare,diff,let,letsimp,pi,sin,solve,trigsimp] [assume,constant,cos,declare,diff,let,letsimp,ratsimp,sin,solve] [assume,constant,cos,declare,diff,let,letsimp,rhs,sin,solve,trigsimp] [assume,constant,cos,declare,diff,let,letsimp,sin,solve,subst,trigsimp] [assume,constant,cos,declare,diff,let,letsimp,sin,solve,trigsimp] [assume,constant,cos,declare,diff,let,letsimp,sin,solve] [assume,constant,cos,declare,diff,let,letsimp,sin,subst,trigsimp] [block,display2d,false,go,let,letsimp,matchdeclare,ratexpand,sin] [constant,cos,declare,diff,let,letsimp,sin,solve] [cos,cot,declare,defrule,evfun,false,let,letsimp,matchdeclare,sin,tan,tellsimp,trigexpand,trigsimp,true] [cos,cot,declare,evfun,false,let,letsimp,matchdeclare,sin,tan,tellsimp,trigexpand,trigsimp,true] [cos,cot,declare,evfun,let,letsimp,matchdeclare,sin,tan,tellsimp,trigexpand,trigsimp,true] [cos,cot,let,letsimp,matchdeclare,sin,tan,true] [determinant,let,letsimp,matrix,ratsimp,scanmap,solve] [determinant,let,letsimp,matrix,simp,solve] [diff,let,letsimp,ode2,solve,subst] [let,letsimp,solve] [let,letsimp] [letsimp]

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