~v200 200 ~w38 0 486 784 400 500 0 169 ~f? 10 9 8
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"Times" 12 ? ? 5 0 c n 106 1 0 0 k 468
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-2 0 26177 26178 26115 26178 1 1 1 1 0 0
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~Q
]|Expr|[#b @`bb#_b#_b#_})3# b'4" *|: ;bP8&c0!*          ;bP9&c0!) |
  |     `f#};bP<mutua ricorsione}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
0 ~V?c3 ('N)~p0
1 ~V?f0 (T)~p0
1 ~V?f0 (S)~p0
1 ~V?f0 (Y')~p0
1 ~V?f0 (X')~p0
1 ~V?f0 (K)~p0
1 ~V?f0 (Y)~p0
1 ~V?f0 (X)~p0
1 ~V?f0 (H)~p0
1 ~V?f0 (g)~p0
1 ~V?f0 (f)~p0
1 ~Q
]|Expr|[#b @`bb#_b#_b#_})!# b$L" *|: ;bP8&c0!*Trigonometry|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
1 ~V?c1 ('p)~p0
2 ~V?f256 (sin)~p0
2 ~V?f257 (cos)~p0
2 ~V?f258 (tan)~p0
2 ~V?f261 (sec)~p0
2 ~V?f260 (csc)~p0
2 ~V?f262 (cot)~p0
2 ~V?f272 (arcsin)~p0
2 ~V?f273 (arccos)~p0
2 ~Q
]|Expr|[#b @`bb#_b#_b#_})!# b$L" *|: ;bP8&c0!*Hyperbolic|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
1 ~V?f293 (sech)~p0
2 ~V?f288 (sinh)~p0
2 ~V?f289 (cosh)~p0
2 ~V?f290 (tanh)~p0
2 ~Q
]|Expr|[#b @`bb#_b#_b#_})%# b$L" *|: ;bP8&c0!*Logarithms ,F Powers|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
1 ~V?f32 (log)~p0
2 ~V?f307 (ln)~p0
2 ~V?f291 (exp)~p0
2 ~V?c2 (e)~p0
2 ~Q
]|Expr|[#b @`bb#_b#_b#_})## b$L" *|: ;bP8&c0!*Standard Rules|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
1 ~Q
]|Expr|[#b @`bb#_b#_b#_})%# b$L" *|: ;bP8&c0!*Logarithms ,F Powers|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
2 ~Ht(?x^(-?y)):(1/?x^?y)~p0
3 ~Hs(exp(?z)):(e^?z)~p0
3 ~Hs(e^(ln(?x))):(?x)~p0
3 ~Hs(10^(log(?x))):(?x)~p0
3 ~Hs(?y^(log_?y(?x))):(?x)~p0
3 ~Hs(ln(e^?x)):(?x)~p0
3 ~Hs(log(10^?x)):(?x)~p0
3 ~Hs(log_?y(?y^?x)):(?x)~p0
3 ~He(ln(?u*?v)):(ln(?u)+ln(?v))~p0
3 ~He(log(?u*?v)):(log(?u)+log(?v))~p0
3 ~He(log_?y(?u*?v)):(log_?y(?u)+log_?y(?v))~p0
3 ~He(ln(?u/?v)):(ln(?u)-ln(?v))~p0
3 ~He(log(?u/?v)):(log(?u)-log(?v))~p0
3 ~He(log_?y(?u/?v)):(log_?y(?u)-log_?y(?v))~p0
3 ~He(ln(?u^?v)):(?v*ln(?u))~p0
3 ~He(log(?u^?v)):(?v*log(?u))~p0
3 ~He(log_?y(?u^?v)):(?v*log_?y(?u))~p0
3 ~He(ln(sqrt(?u))):(1/2*ln(?u))~p0
3 ~He(log(sqrt(?u))):(1/2*log(?u))~p0
3 ~He(log_?y(sqrt(?u))):(1/2*log_?y(?u))~p0
3 ~Q
]|Expr|[#b @`bb#_b#_b#_})!# b$L" *|: ;bP8&c0!*Trigonometry|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
2 ~Q
]|Expr|[#b @`bb#_b#_b#_})+# b$L" *|: ;bP8&c0!*Simplify ,M negation|
  | and common zeros}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
3 ~Hs(sin(-?x)):(-sin(?x))~p0
4 ~Hs(cos(-?x)):(cos(?x))~p0
4 ~Hs(tan(-?x)):(-tan(?x))~p0
4 ~Hs(sin('p)):(0)~p0
4 ~Hs(sin(?n*'p)):(0)~p0
4 ~Hs(cos(1/2*'p)):(0)~p0
4 ~Hs(cos(?n/2*'p)):(0)~p0
4 ~Q
]|Expr|[#b @`bb#_b#_b#_})'# b$L" *|: ;bP8&c0!*Transform to basic|
  | types}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
3 ~Ht(tan(?x)):((sin(?x))/(cos(?x)))~p0
4 ~Ht(csc(?x)):(1/(sin(?x)))~p0
4 ~Ht(sin(?x)):(1/(csc(?x)))~p0
4 ~Ht(sec(?x)):(1/(cos(?x)))~p0
4 ~Ht(cos(?x)):(1/(sec(?x)))~p0
4 ~Ht(cot(?x)):((cos(?x))/(sin(?x)))~p0
4 ~Q
]|Expr|[#b @`bb#_b#_b#_})## b$L" *|: ;bP8&c0!*Trig Addition|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
3 ~Ht(cos(?x+?y)):(cos(?x)*cos(?y)-sin(?x)*sin(?y))~p0
4 ~Ht(sin(?x+?y)):(cos(?x)*sin(?y)+sin(?x)*cos(?y))~p0
4 ~Ht(cos(2*?x)):(2*(cos(?x))^2-1)~p0
4 ~Ht(sin(2*?x)):(2*cos(?x)*sin(?x))~p0
4 ~Ht(sin(?n*?x)):(cos((?n-1)*?x)*sin(?x)+cos(?x)*sin((?n-1)*?x))~p0
4 ~Ht(cos(?n*?x)):(cos(?x)*cos((?n-1)*?x)-sin(?x)*sin((?n-1)*?x))~p0
4 ~Q
]|Expr|[#b @`bb#_b#_b#_})## b$L" *|: ;bP8&c0!*Trig Subtraction|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
3 ~Ht(cos(?x-?y)):(cos(?x)*cos(?y)+sin(?x)*sin(?y))~p0
4 ~Ht(sin(?x-?y)):(sin(?x)*cos(?y)-cos(?x)*sin(?y))~p0
4 ~Q
]|Expr|[#b @`bb#_b#_b#_})/# b$L" *|: ;bP8&c0!*Transform ,M into|
  | another flavor of trig function}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
3 ~Ht((sin(?x))^2):(1-(cos(?x))^2)~p0
4 ~Ht((cos(?x))^2):(1-(sin(?x))^2)~p0
4 ~Ht((tan(?x))^2):((sec(?x))^2-1)~p0
4 ~Ht((sec(?x))^2):((tan(?x))^2+1)~p0
4 ~Ht((csc(?x))^2):((cot(?x))^2+1)~p0
4 ~Ht((cot(?x))^2):((csc(?x))^2-1)~p0
4 ~Hs((sin(?x))^2+(cos(?x))^2):(1)~p0
4 ~Q
]|Expr|[#b @`bb#_b#_b#_})b @# b%4" *|: ;bP8&c0!*substituting |
  |z,]tan,Hx,O2,I into a rational function in sin,Hx,I and cos,H|
  |x,I}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
3 ~Hs(cos(2*sin(?a+?b)*?z)):((1-?z^2)/(1+?z^2))~p0
4 ~Hs(sin(2*sin(?a+?b)*?z)):(2*?z/(1+?z^2))~p0
4 ~Q
]|Expr|[#b @`bb#_b#_b#_})## b$L" *|: ;bP8&c0!*Other rules|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
3 ~Ht((cos(?x))^2):(1/2*(cos(2*?x)+1))~p0
4 ~Ht((sin(?x))^2):(1/2*(-cos(2*?x)+1))~p0
4 ~Ht(cos(?x)*sin(?x)):(1/2*sin(2*?x))~p0
4 ~Hs(sin(arccos(?x))):(sqrt(-?x^2+1))~p0
4 ~Hs(cos(arcsin(?x))):(sqrt(-?x^2+1))~p0
4 ~Q
]|Expr|[#b @`bb#_b#_b#_})!# b!T" *|: ;bP8&c0!*Hyperbolic|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
2 ~Q
]|Expr|[#b @`bb#_b#_b#_})+# b$L" *|: ;bP8&c0!*Simplify ,M negation|
  | and common zeros}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
3 ~Hs(sinh(-?x)):(-sinh(?x))~p0
4 ~Hs(cosh(-?x)):(cosh(?x))~p0
4 ~Hs(tanh(-?x)):(-tanh(?x))~p0
4 ~Q
]|Expr|[#b @`bb#_b#_b#_})'# b$L" *|: ;bP8&c0!*Transform into |
  |other types}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
3 ~Ht((sinh(?x))^2):((cosh(?x))^2-1)~p0
4 ~Ht((cosh(?x))^2):(1+(sinh(?x))^2)~p0
4 ~Ht((tanh(?x))^2):(1-(sech(?x))^2)~p0
4 ~Ht(sinh(?x)):((e^?x-e^(-?x))/2)~p0
4 ~Ht(cosh(?x)):((e^?x+e^(-?x))/2)~p0
4 ~Ht(tanh(?x)):((e^?x-e^(-?x))/(e^?x+e^(-?x)))~p0
4 ~Ht(tanh(?x)):((sinh(?x))/(cosh(?x)))~p0
4 ~Hs((cosh(?x))^2-(sinh(?x))^2):(1)~p0
4 ~Hs(-(cosh(?x))^2+(sinh(?x))^2):(-1)~p0
4 ~Q
]|Expr|[#b @`bb#_b#_b#_})%# b$L" *|: ;bP8&c0!*Other hyperbolic|
  | rules}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
3 ~Ht((cosh(?x))^2):(1/2*(cosh(2*?x)+1))~p0
4 ~Ht((sinh(?x))^2):(1/2*(cosh(2*?x)-1))~p0
4 ~Ht(sinh(2*?x)):(2*cosh(?x)*sinh(?x))~p0
4 ~Ht(cosh(2*?x)):(2*(cosh(?x))^2-1)~p0
4 ~Q
]|Expr|[#b @`bb#_b#_b#_})## b$L" *|: ;bP8&c0!*Integration Rules|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
2 ~Q
]|Expr|[#b @`bb#_b#_b#_}))# b$8" *|: ;bP8&c0!*after Partial Fraction|
  | Decomposition integration}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
3 ~Hs(ln(?x+?b*i)):((-i*sin(?a+?b)*?x/?b+1/2*ln(?x^2+?b^2))+1/2*~
 'p*i)~p0
4 ~Hs(ln(?x+i)):((-i*sin(?a+?b)*?x+1/2*ln(?x^2+1))+1/2*'p*i)~p0
4 ~Hs(ln(?x-?b*i)):((i*sin(?a+?b)*?x/?b+1/2*ln(?x^2+?b^2))+1/2*~
 'p*i)~p0
4 ~Hs(ln(?x-i)):((i*sin(?a+?b)*?x+1/2*ln(?x^2+1))+1/2*'p*i)~p0
4 ~Q
]|Expr|[#b @`bb#_b#_b#_})%# b$4" *|: ;bP8&c0!*Derivatives of |
  |Integrals}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p1
3 ~Ht(Diff(?x)*(Integral(?y*d*?x))):(?y)~p0
4 ~V?c4 (i)~p0
1 ~V?d16 (d)~p0
1 ~V?v0 (k)~p0
1 ~V?c0 (n)~p0
1 ~V?c0 (c)~p0
1 ~V?c0 (b)~p0
1 ~V?c0 (a)~p0
1 ~V?v0 (z)~p0
1 ~V?v0 (y)~p0
1 ~V?v0 (x)~p0
1 ~Q
]|Expr|[#b @`bb#_b#_b#_}) # b'4}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
0 ~Q
]|Expr|[#b @`bb#_b#_b#_}`fb"#})-# b'4" *|: ;bP8&c0!*funzioni |
  |che attuano il passo della ricorsione}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
0 ~A(f(?x,?y,?n)=3*?x-?y+?n)~p0
1 ~d~A(g(?x,?y,?n)=-3*?x+5*?y+8^?n)~p0
1 ~d~Q
]|Expr|[#b @`bb#_b#_b#_}`fb#@})## b!@" *|: ;bP8&c0!*valori iniziali|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
0 ~A(a=2)~p0
255 ~d~A(b=0)~p0
255 ~d~Q
]|Expr|[#b @`bb#_b#_b#_}`fb#@})+# b$&" *|: ;bP8&c0!*indice da|
  | considerare per la valutazione}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
0 ~A(n=3)~p0
255 ~d~Q
]|Expr|[#b @`bb#_b#_b#_}`fb#C})1# b%\" *|: ;bP8&c0!*definizione|
  | diretta ,Hsi affrontano tutti gli indici insieme,I|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
0 ~A(H(?n)=Conditional((a,b),(?n=0);(f((H(?n-1))_1,(H(?n-1))_2,~
 ?n-1),g((H(?n-1))_1,(H(?n-1))_2,?n-1)),(?n>0)))~p0
1 ~d~Q
]|Expr|[#b @`bb#_b#_b#_}`fb#C})!# b'4" *|: ;bP8&c0!*esempio|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
1 ~A(H(n))~p0
2 ~A(H(n)=(109,-183))~p0
3 ~sb/^!!   }
$&!  c#T"!c#L"_c/__c/__}
^ _~Q
]|Expr|[#b @`bb#_b#_b#_}`fb!D})5# b'4" *|: ;bP8&c0!*definizione|
  | mediata ,Hsi usano le funzioni ausiliarie X e Y,I|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
0 ~A(K(?n)=Conditional((a,b),(?n=0);(f(X(?n-1),Y(?n-1),?n-1),g(~
 X(?n-1),Y(?n-1),?n-1)),(?n>0)))~p0
1 ~d~A(X(?n)=(K(?n))_1)~p0
2 ~d~A(Y(?n)=(K(?n))_2)~p0
255 ~d~Q
]|Expr|[#b @`bb#_b#_b#_}`fb!D})!# b'4" *|: ;bP8&c0!*esempio|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
1 ~A(K(n))~p0
1 ~A(K(n)=(109,-183))~p0
2 ~sb/^!!   }
$&!  c#T"!c#L"_c/__c/__}
^ _~A(X(n))~p0
1 ~A(X(n)=109)~p0
2 ~sb/^!!   }
$&!  c#T"!c#L"_c/__c/__}
^ _~A(Y(n))~p0
1 ~A(Y(n)=-183)~p0
2 ~sb/^!!   }
$&!  c#T"!c#L"_c/__c/__}
^ _~Q
]|Expr|[#b @`bb#_b#_b#_}`fb#L})0# b'4" *|: ;bP8&c0!*il procedimento|
  | seguente invece non funziona in MathView|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
0 ~A(X'(?n)=f(X'(?n-1),Y'(?n-1),?n-1))~p0
1 ~A(Y'(?n)=g(X'(?n-1),Y'(?n-1),?n-1))~p0
1 ~d~Q
]|Expr|[#b @`bb#_b#_b#_}`fb#L})1# b'4" *|: ;bP8&c0!*prova a effettuare|
  | ,Bcalculate,B sui seguenti input ,Z}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
2 ~A(X'(5))~p0
3 ~A(Y'(5))~p0
3 ~A(X'(5),Y'(5))~p0
3 ~Q
]|Expr|[#b @`bb#_b#_b#_}) # b'4}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
3 ~A(S(?n)=(X'(?n),Y'(?n)))~p0
4 ~d~A(S(5))~p0
4 ~Q
]|Expr|[#b @`bb#_b#_b#_}`f#})b!+# b'4" *|: ;bP8&c0 JPertanto |
  |&c0!*,Z  le due componenti della ricorsione devono essere coinvolte|
  | nella stessa funzione sin dalla definizione iniziale,[ si pu|
  |/R solo delegare a funzioni ausiliarie ,Hfunzioni X e Y,I  il|
  | processo di proiezione sulle componenti ,N|
  |}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
1 ~Q
]|Expr|[#b @`bb#_b#_b#_}`fb#@})-# b'4" *|: ;bP8&c0!*algoritmo|
  | iterativo   ,H tail recursion ,I}& b!( b"0 b#8 b$@ b%H b&P!WW}]|[~p0
0 ~A(T(?s,?x,?y,?n)=Conditional((?x,?y),(?s=?n);T(?s+1,f(?x,?y,~
 ?s),g(?x,?y,?s),?n),(?s<?n)))~p0
1 ~d~A(T(0,a,b,n))~p0
2 ~A(T(0,a,b,n)=(109,-183))~p0
3 ~sb/^!!   }
$&!  c#T"!c#L"_c/__c/__}
^ _~A(T(0,a,b,10))~p0
2 ~A(T(0,a,b,10)=(-51080211,332206616))~p0
3 ~sb/^!!   }
$&!  c#T"!c#L"_c/__c/__}
^ _~c13
148 -1 147 -1 143 -1 128 -1 140 -1 131 -1 141 -1 130 -1 145 -1 9 -1 137 -1 11 -1 138 -1 10 -1 ~c17
155 -1 154 -1 143 -1 128 -1 140 -1 131 -1 141 -1 130 -1 150 -1 6 -1 151 -1 8 -1 152 -1 7 -1 137 -1 11 -1 138 -1 10 -1 ~c17
157 -1 156 -1 143 -1 128 -1 140 -1 131 -1 141 -1 130 -1 150 -1 6 -1 151 -1 8 -1 152 -1 7 -1 137 -1 11 -1 138 -1 10 -1 ~c17
159 -1 158 -1 143 -1 128 -1 140 -1 131 -1 141 -1 130 -1 150 -1 6 -1 151 -1 8 -1 152 -1 7 -1 137 -1 11 -1 138 -1 10 -1 ~c13
174 -1 173 -1 140 -1 131 -1 141 -1 130 -1 143 -1 128 -1 137 -1 11 -1 138 -1 10 -1 172 -1 2 -1 ~c11
176 -1 175 -1 140 -1 131 -1 141 -1 130 -1 137 -1 11 -1 138 -1 10 -1 172 -1 2 -1 ~e