#
# WORKSHEET#20
#
#
# Some special functions
#
#
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> ?GAMMA
#
#
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> GAMMA(s):=int(exp(-t)*t^(s-1),t=0..infinity);
infinity
/
| (s - 1)
GAMMA(s) := | exp(- t) t dt
|
/
0
> GAMMA(0);
Error, (in GAMMA) singularity encountered
--------------------------------------------------------------------------------
> for n from 1 to 10 do GAMMA(n) od;
1
1
2
6
24
120
720
5040
40320
362880
--------------------------------------------------------------------------------
> n:='n':
--------------------------------------------------------------------------------
> for n from 1 to 10 do A[n]:=GAMMA(n+1)/GAMMA(n) od;
A[1] := 1
A[2] := 2
A[3] := 3
A[4] := 4
A[5] := 5
A[6] := 6
A[7] := 7
A[8] := 8
A[9] := 9
A[10] := 10
--------------------------------------------------------------------------------
> GAMMA(1/2);
1/2
Pi
--------------------------------------------------------------------------------
> n:='n': A[n]:='A[n]': for n from 1 to 10 do \
A[n]:=GAMMA(n+(1/2))/GAMMA(n-(1/2)) od;
A[1] := 1/2
A[2] := 3/2
A[3] := 5/2
A[4] := 7/2
A[5] := 9/2
A[6] := 11/2
A[7] := 13/2
A[8] := 15/2
A[9] := 17/2
A[10] := 19/2
--------------------------------------------------------------------------------
> simplify(GAMMA(s+1)/GAMMA(s)-s);\
0
> n:='n': A[n]:='A[n]':
> ?Si;
--------------------------------------------------------------------------------
> Si(x):=Int((sin(t))/t,t=0..x);
x
/
| sin(t)
Si(x) := | ------ dt
| t
/
0
--------------------------------------------------------------------------------
> Si(0);
0
--------------------------------------------------------------------------------
> Si(infinity);
1/2 Pi
--------------------------------------------------------------------------------
> plot(sin(t)/t,t=0..100);
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#
** Maple V Graphics **
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> for k from 1 to 10 do X[k]:= evalf(Int(sin(t)/t,t=(k-1)*Pi..k*Pi)) od;\
X[1] := 1.851937052
X[2] := -.4337854758
X[3] := .2566102228
X[4] := -.1826005734
X[5] := .1418036205
X[6] := -.1159308846
X[7] := .09805155598
X[8] := -.08495423246
X[9] := .07494560683
X[10] := -.06704781224
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> sum('X[k]','k=1..10');
1.539029081
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> evalf(Pi/2);
1.570796327
--------------------------------------------------------------------------------
