### Related

alpha_1:1;

Fr_1:2^0.5;

x0_1:%i*0.577;

Calculate

##### assoc_legendre_q-diff-expand-legendre_q

a:-%i/(3^(0.5));

a:%ig(x):= ((x^2-1)^(...

g(x):= legendre_q(5,x);

Calculate

Fr_0:2^0.5;

x0_0:%i;

alpha_0:0;

Calculate

alpha_1:1;

Fr_1:2^0.5;

x0_1:%i*0.577;

Calculate

##### assoc_legendre_q-diff-float-legendre_q-numer-plot2d-realpart

x0:%i*0.447;

F1(x):=((1-x*x)^(-1))...

DiffF1(x):= ''(diff(F...

Calculate

f(x):= x+1;

a:%i/sqrt(3);

expand(f(a));

Calculate

alpha_1:1;

Fr_1:2^0.5;

x0_1:%i*0.577;

Calculate

Fr_0:2^0.5;

x0_0:%i;

alpha_0:0;

Calculate

Fr:sqrt(2);

Alpha:1;

eta:1;

Calculate

Fr_2:2^0.5;

alpha_2:2;

x0_2:%i*0.447;

Calculate

### assoc_legendre_q

Run Example
```(%i1)a:-%i/(3^(0.5));
(%o1)                        - 0.57735026918963 %i
(%i2) a:%ig(x):= ((x^2-1)^(-1/2))*assoc_legendre_q(8,-1,x);
- 1
---
2      2
(%o2)          %ig(x) := (x  - 1)    assoc_legendre_q(8, - 1, x)
(%i3) g(x):= legendre_q(5,x);
(%o3)                      g(x) := legendre_q(5, x)
(%i4) expand(g(a));
- 1
---
2      2     - 1    5
(%o4) (63 (%ig(x) := (x  - 1)    "Q"   (x))
8
- 1
---
2      2     - 1
%ig(x) := (x  - 1)    "Q"   (x)
8
log(- -------------------------------------
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
1
- -------------------------------------))/16
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
- 1
---
2      2     - 1    3
- (35 (%ig(x) := (x  - 1)    "Q"   (x))
8
- 1
---
2      2     - 1
%ig(x) := (x  - 1)    "Q"   (x)
8
log(- -------------------------------------
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
1
- -------------------------------------))/8
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
- 1
---
2      2     - 1
+ (15 (%ig(x) := (x  - 1)    "Q"   (x))
8
- 1
---
2      2     - 1
%ig(x) := (x  - 1)    "Q"   (x)
8
log(- -------------------------------------
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
1
- -------------------------------------))/16
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
- 1
---
2      2     - 1    4
63 (%ig(x) := (x  - 1)    "Q"   (x))
8
- -------------------------------------
8
- 1
---
2      2     - 1    2
49 (%ig(x) := (x  - 1)    "Q"   (x))
8          8
+ ------------------------------------- - --
8                     15
(%i5) g1(x):=''(diff(g(x),x));
x + 1   6        5             x + 1   4        3
(%o5)/R/ g1(x) := (315 log(- -----) x  - 630 x  - 525 log(- -----) x  + 840 x
x - 1                          x - 1
x + 1   2                    x + 1        2
+ 225 log(- -----) x  - 226 x - 15 log(- -----))/(16 x  - 16)
x - 1                        x - 1
(%i6) expand(g1(a));
- 1
---
2      2     - 1    6
(%o6) (315 (%ig(x) := (x  - 1)    "Q"   (x))
8
- 1
---
2      2     - 1
%ig(x) := (x  - 1)    "Q"   (x)
8
log(- -------------------------------------
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
1
- -------------------------------------))
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
- 1
---
2      2     - 1    2
/(16 (%ig(x) := (x  - 1)    "Q"   (x))  - 16)
8
- 1
---
2      2     - 1    4
- (525 (%ig(x) := (x  - 1)    "Q"   (x))
8
- 1
---
2      2     - 1
%ig(x) := (x  - 1)    "Q"   (x)
8
log(- -------------------------------------
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
1
- -------------------------------------))
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
- 1
---
2      2     - 1    2
/(16 (%ig(x) := (x  - 1)    "Q"   (x))  - 16)
8
- 1
---
2      2     - 1    2
+ (225 (%ig(x) := (x  - 1)    "Q"   (x))
8
- 1
---
2      2     - 1
%ig(x) := (x  - 1)    "Q"   (x)
8
log(- -------------------------------------
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
1
- -------------------------------------))
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
- 1
---
2      2     - 1    2
/(16 (%ig(x) := (x  - 1)    "Q"   (x))  - 16)
8
- 1
---
2      2     - 1
%ig(x) := (x  - 1)    "Q"   (x)
8
- (15 log(- -------------------------------------
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
1
- -------------------------------------))
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
- 1
---
2      2     - 1    2
/(16 (%ig(x) := (x  - 1)    "Q"   (x))  - 16)
8
- 1
---
2      2     - 1    5
630 (%ig(x) := (x  - 1)    "Q"   (x))
8
- ------------------------------------------
- 1
---
2      2     - 1    2
16 (%ig(x) := (x  - 1)    "Q"   (x))  - 16
8
- 1
---
2      2     - 1    3
840 (%ig(x) := (x  - 1)    "Q"   (x))
8
+ ------------------------------------------
- 1
---
2      2     - 1    2
16 (%ig(x) := (x  - 1)    "Q"   (x))  - 16
8
- 1
---
2      2     - 1
226 %ig(x) := (x  - 1)    "Q"   (x)
8
- ------------------------------------------
- 1
---
2      2     - 1    2
16 (%ig(x) := (x  - 1)    "Q"   (x))  - 16
8
(%i7) A1:(4.0)/(7.0);
(%o7)                          0.57142857142857
(%i8) B:(125*25)/(56*21);
3125
(%o8)                                ----
1176
(%i9) E:(125)/(1512);
125
(%o9)                                ----
1512
(%i10) F:(-25)/(196);
25
(%o10)                               - ---
196
(%i11) Exp1:B+a*A1*E*2.0*a/((1-a*a)^2)-F-E/(1-a*a);
125
(%o11) - ---------------------------------------------
- 1
---
2      2     - 1    2
1512 (1 - (%ig(x) := (x  - 1)    "Q"   (x)) )
8
- 1
---
2      2     - 1    2
0.09448223733938 (%ig(x) := (x  - 1)    "Q"   (x))
8          3275
+ --------------------------------------------------- + ----
- 1                      1176
---
2      2     - 1    2 2
(1 - (%ig(x) := (x  - 1)    "Q"   (x)) )
8
(%i12) Exp1:9/200+27*3/50;
333
(%o12)                                ---
200
(%i13) Exp2: g(a)-a*A1*g1(a);

rat: replaced 0.57142857142857 by 4/7 = 0.57142857142857
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) + 1
8
(%o13)/R/ - (12285 log(- -------------------------------------)
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
- 1                                        - 1
---                                        ---
2      2     - 1    7                      2      2     - 1    6
(%ig(x) := (x  - 1)    "Q"   (x))  - 24570 (%ig(x) := (x  - 1)    "Q"   (x))
8                                          8
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) + 1
8
- 17535 log(- -------------------------------------)
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
- 1                                        - 1
---                                        ---
2      2     - 1    5                      2      2     - 1    4
(%ig(x) := (x  - 1)    "Q"   (x))  + 26880 (%ig(x) := (x  - 1)    "Q"   (x))
8                                          8
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) + 1
8
+ 4575 log(- -------------------------------------)
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
- 1                                       - 1
---                                       ---
2      2     - 1    3                     2      2     - 1    2
(%ig(x) := (x  - 1)    "Q"   (x))  - 2374 (%ig(x) := (x  - 1)    "Q"   (x))
8                                         8
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) + 1
8
+ 675 log(- -------------------------------------)
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 896)
8
- 1
---
2      2     - 1    2
/(1680 (%ig(x) := (x  - 1)    "Q"   (x))  - 1680)
8
(%i14) Exp2: g(a)+ a*(12/30)*g1(a);
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) + 1
8
(%o14)/R/ (2835 log(- -------------------------------------)
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
- 1                                       - 1
---                                       ---
2      2     - 1    7                     2      2     - 1    6
(%ig(x) := (x  - 1)    "Q"   (x))  - 5670 (%ig(x) := (x  - 1)    "Q"   (x))
8                                         8
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) + 1
8
- 5145 log(- -------------------------------------)
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
- 1                                       - 1
---                                       ---
2      2     - 1    5                     2      2     - 1    4
(%ig(x) := (x  - 1)    "Q"   (x))  + 8400 (%ig(x) := (x  - 1)    "Q"   (x))
8                                         8
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) + 1
8
+ 2625 log(- -------------------------------------)
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
- 1                                       - 1
---                                       ---
2      2     - 1    3                     2      2     - 1    2
(%ig(x) := (x  - 1)    "Q"   (x))  - 2954 (%ig(x) := (x  - 1)    "Q"   (x))
8                                         8
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) + 1
8
- 315 log(- -------------------------------------)
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) + 128)
8
- 1
---
2      2     - 1    2
/(240 (%ig(x) := (x  - 1)    "Q"   (x))  - 240)
8
(%i15) C2:expand(Exp1/Exp2);
- 1
---
2      2     - 1    2
(%o15) (1998 (%ig(x) := (x  - 1)    "Q"   (x)) )
8
- 1
---
2      2     - 1    7
/(14175 (%ig(x) := (x  - 1)    "Q"   (x))
8
- 1
---
2      2     - 1
%ig(x) := (x  - 1)    "Q"   (x)
8
log(- -------------------------------------
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
1
- -------------------------------------)
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
- 1
---
2      2     - 1    5
- 25725 (%ig(x) := (x  - 1)    "Q"   (x))
8
- 1
---
2      2     - 1
%ig(x) := (x  - 1)    "Q"   (x)
8
log(- -------------------------------------
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
1
- -------------------------------------)
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
- 1
---
2      2     - 1    3
+ 13125 (%ig(x) := (x  - 1)    "Q"   (x))
8
- 1
---
2      2     - 1
%ig(x) := (x  - 1)    "Q"   (x)
8
log(- -------------------------------------
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
1
- -------------------------------------)
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
- 1
---
2      2     - 1
- 1575 (%ig(x) := (x  - 1)    "Q"   (x))
8
- 1
---
2      2     - 1
%ig(x) := (x  - 1)    "Q"   (x)
8
log(- -------------------------------------
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
1
- -------------------------------------)
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
- 1
---
2      2     - 1    6
- 28350 (%ig(x) := (x  - 1)    "Q"   (x))
8
- 1
---
2      2     - 1    4
+ 42000 (%ig(x) := (x  - 1)    "Q"   (x))
8
- 1
---
2      2     - 1    2
- 14770 (%ig(x) := (x  - 1)    "Q"   (x))  + 640)
8
- 1
---
2      2     - 1    7
- 1998/(14175 (%ig(x) := (x  - 1)    "Q"   (x))
8
- 1
---
2      2     - 1
%ig(x) := (x  - 1)    "Q"   (x)
8
log(- -------------------------------------
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
1
- -------------------------------------)
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
- 1
---
2      2     - 1    5
- 25725 (%ig(x) := (x  - 1)    "Q"   (x))
8
- 1
---
2      2     - 1
%ig(x) := (x  - 1)    "Q"   (x)
8
log(- -------------------------------------
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
1
- -------------------------------------)
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
- 1
---
2      2     - 1    3
+ 13125 (%ig(x) := (x  - 1)    "Q"   (x))
8
- 1
---
2      2     - 1
%ig(x) := (x  - 1)    "Q"   (x)
8
log(- -------------------------------------
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
1
- -------------------------------------)
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
- 1
---
2      2     - 1
- 1575 (%ig(x) := (x  - 1)    "Q"   (x))
8
- 1
---
2      2     - 1
%ig(x) := (x  - 1)    "Q"   (x)
8
log(- -------------------------------------
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
1
- -------------------------------------)
- 1
---
2      2     - 1
(%ig(x) := (x  - 1)    "Q"   (x)) - 1
8
- 1
---
2      2     - 1    6
- 28350 (%ig(x) := (x  - 1)    "Q"   (x))
8
- 1
---
2      2     - 1    4
+ 42000 (%ig(x) := (x  - 1)    "Q"   (x))
8
- 1
---
2      2     - 1    2
- 14770 (%ig(x) := (x  - 1)    "Q"   (x))  + 640)
8
(%i16) ```
Run Example
```load(interpol);
(%o1)         /usr/share/maxima/5.21.1/share/numeric/interpol.mac
(%i2) alpha_1:1;
(%o2)                                  1
(%i3) Fr_1:2^0.5;
(%o3)                          1.414213562373095
(%i4) x0_1:%i*0.577;
(%o4)                              0.577 %i
(%i5) beta_1:(2*(alpha_1+1)/(3+2*alpha_1));
4
(%o5)                                  -
5
(%i6) A_1:(3+2*alpha_1)*(3+2*alpha_1)*(3+2*alpha_1)/((4*alpha_1+8)*(2*alpha_1+5));
125
(%o6)                                 ---
84
(%i7) C2_1:-0.403;
(%o7)                               - 0.403
(%i8) H1p_1(x):=(125.0/1512)/(1-x*x) - 25.0/196;
125.0
-----
1512     25.0
(%o8)                     H1p_1(x) := ------- - ----
1 - x x   196
(%i9) DiffH1p_1(x):= ''(diff(H1p_1(x), x));
0.16534391534392 x
(%o9)                 DiffH1p_1(x) := ------------------
2 2
(1 - x )
(%i10) F1_1(x):=(1-x*x)^(-alpha_1/2)*assoc_legendre_q(8,-1,x);
- alpha_1
---------
2
(%o10)     F1_1(x) := (1 - x x)          assoc_legendre_q(8, - 1, x)
(%i11) DiffF1_1(x):= ''(diff(F1_1(x), x));
x + 1   10          9
(%o11)/R/ DiffF1_1(x) := - (45045 log(- -----) x   - 90090 x
x - 1
x + 1   8           7                x + 1   6           5
- 135135 log(- -----) x  + 240240 x  + 145530 log(- -----) x  - 218988 x
x - 1                                x - 1
x + 1   4          3               x + 1   2
- 66150 log(- -----) x  + 76464 x  + 11025 log(- -----) x  - 7562 x
x - 1                              x - 1
x + 1          4         2
- 315 log(- -----))/(2304 x  - 4608 x  + 2304)
x - 1
(%i12) H1_1(x)=C2_1*F1_1(x) +H1p_1(x);

rat: replaced -0.403 by -403/1000 = -0.403

rat: replaced -0.1275510204082 by -25/196 = -0.1275510204082

rat: replaced 0.082671957671958 by 125/1512 = 0.082671957671958
x + 1   9               8
(%o12)/R/ H1_1(x) = (1906079175 log(- -----) x  - 3812158350 x
x - 1
x + 1   7               6                    x + 1   5
- 4574590020 log(- -----) x  + 7878460590 x  + 3694861170 log(- -----) x
x - 1                                        x - 1
4                    x + 1   3              2
- 5102427330 x  - 1119654900 log(- -----) x  + 845310978 x
x - 1
x + 1                             2
+ 93304575 log(- -----) x + 41335552)/(1693440000 x  - 1693440000)
x - 1
(%i13) DiffH1_1(x):= DiffF1_1(x)*C2_1+DiffH1p_1(x);
(%o13)          DiffH1_1(x) := DiffF1_1(x) C2_1 + DiffH1p_1(x)
(%i14) H0_1(x):=1.0/(Fr_1*Fr_1) - 1.0/(4*alpha_1+4) + x^2/(4*alpha_1+4);
2
1.0           1.0             x
(%o14)       H0_1(x) := --------- - ------------- + -------------
Fr_1 Fr_1   4 alpha_1 + 4   4 alpha_1 + 4
(%i15) DiffH0_1(x):= ''(diff(H0_1(x), x));
x
(%o15)                         DiffH0_1(x) := -
4
(%i16) B_1:(3+2*alpha_1)*(3+2*alpha_1)*(3+2*alpha_1)/((8*alpha_1+20)*(alpha_1+1));
125
(%o16)                                ---
56
(%i17) Fr_2:2^0.5;
(%o17)                         1.414213562373095
(%i18) U1_Al1(zeta):=realpart( (-beta_1/2)*( DiffH1_1(zeta*x0_1)*x0_1 -A_1*DiffH0_1(zeta)/H0_1(zeta)));
- beta_1
(%o18) U1_Al1(zeta) := realpart(--------
2
A_1 DiffH0_1(zeta)
(DiffH1_1(zeta x0_1) x0_1 - ------------------))
H0_1(zeta)
(%i19) Phi1_Al1(y):= B_1/H0_1(y);
B_1
(%o19)                      Phi1_Al1(y) := -------
H0_1(y)
(%i20) Phi2_Al1(y):= -Phi1_Al1(y)/(2.0*H0_1(y))*(125.0/56.0 +  realpart(C2_1*F1_1(y*x0_1)+H1p_1(y*x0_1))) ;
- Phi1_Al1(y)  125.0
(%o20) Phi2_Al1(y) := ------------- (-----
2.0 H0_1(y)   56.0
+ realpart(C2_1 F1_1(y x0_1) + H1p_1(y x0_1)))
(%i21) foo_1(y):= ''(2.0*Phi1_Al1(y)*H0_1(y)*realpart(C2_1*F1_1(y*x0_1)+H1p_1(y*x0_1)));

rat: replaced 0.577 by 577/1000 = 0.577

rat: replaced -0.403 by -403/1000 = -0.403

rat: replaced -0.1275510204082 by -25/196 = -0.1275510204082

rat: replaced 0.082671957671958 by 125/1512 = 0.082671957671958

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786
11
(%o21) foo_1(y) := (4.464285714285714 (741724899620869378121796486996078681 y
2                                         577 y
sqrt(332929 y  + 1000000) (atan2(577 y, - 1000) - atan(-----) - %pi)
1000
9              2
+ 7574782194501581116776447858489798438 y  sqrt(332929 y  + 1000000)
577 y
(atan2(577 y, - 1000) - atan(-----) - %pi)
1000
7              2
+ 29031893253004367050799515694452800000 y  sqrt(332929 y  + 1000000)
577 y
(atan2(577 y, - 1000) - atan(-----) - %pi)
1000
5              2
+ 50769125747525708318584807200000000000 y  sqrt(332929 y  + 1000000)
577 y
(atan2(577 y, - 1000) - atan(-----) - %pi)
1000
3              2
+ 38418615447231297096000000000000000000 y  sqrt(332929 y  + 1000000)
577 y
(atan2(577 y, - 1000) - atan(-----) - %pi)
1000
2
+ 8876601654102000000000000000000000000 y sqrt(332929 y  + 1000000)
577 y
(atan2(577 y, - 1000) - atan(-----) - %pi)
1000
10              2
- 2570970189327103563680403767750706000 y   sqrt(332929 y  + 1000000)
8              2
- 23681651611442049468605076676349388000 y  sqrt(332929 y  + 1000000)
6              2
- 78981841855515216067805804248800000000 y  sqrt(332929 y  + 1000000)
2
- 5715474186240000000000000000000000000 sqrt(332929 y  + 1000000)
2                                                           2
+ y  (16028853347520000000000000000000000000000 sqrt(0.332929 y  + 1)
2
- 60161744582671382560000000000000000000 sqrt(332929 y  + 1000000))
4                                                          2
+ y  (3947525835840000000000000000000000000000 sqrt(0.332929 y  + 1)
2
- 112645908725371144582586400000000000000 sqrt(332929 y  + 1000000))
2
+ 12530880000000000000000000000000000000000 sqrt(0.332929 y  + 1)))
4                2
/(- 30948602552985600000000000000000000000000 y  sqrt(0.332929 y  + 1)
2                2
- 185917133116108800000000000000000000000000 y  sqrt(0.332929 y  + 1)
2
- 279214387200000000000000000000000000000000 sqrt(0.332929 y  + 1))
(%i22)  S_FoP_Al1(y):= ''(integrate( foo_1(y),y,0,1))  ;

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786

rat: replaced 0.332929 by 4117/12366 = 0.33292899886786
(%o22) S_FoP_Al1(y) := 4.464285714285714
1
/
[                                         11              2
I  (741724899620869378121796486996078681 y   sqrt(332929 y  + 1000000)
]
/
0
577 y
(atan2(577 y, - 1000) - atan(-----) - %pi)
1000
9              2
+ 7574782194501581116776447858489798438 y  sqrt(332929 y  + 1000000)
577 y
(atan2(577 y, - 1000) - atan(-----) - %pi)
1000
7              2
+ 29031893253004367050799515694452800000 y  sqrt(332929 y  + 1000000)
577 y
(atan2(577 y, - 1000) - atan(-----) - %pi)
1000
5              2
+ 50769125747525708318584807200000000000 y  sqrt(332929 y  + 1000000)
577 y
(atan2(577 y, - 1000) - atan(-----) - %pi)
1000
3              2
+ 38418615447231297096000000000000000000 y  sqrt(332929 y  + 1000000)
577 y
(atan2(577 y, - 1000) - atan(-----) - %pi)
1000
2
+ 8876601654102000000000000000000000000 y sqrt(332929 y  + 1000000)
577 y
(atan2(577 y, - 1000) - atan(-----) - %pi)
1000
10              2
- 2570970189327103563680403767750706000 y   sqrt(332929 y  + 1000000)
8              2
- 23681651611442049468605076676349388000 y  sqrt(332929 y  + 1000000)
6              2
- 78981841855515216067805804248800000000 y  sqrt(332929 y  + 1000000)
2
- 5715474186240000000000000000000000000 sqrt(332929 y  + 1000000)
2                                                           2
+ y  (16028853347520000000000000000000000000000 sqrt(0.332929 y  + 1)
2
- 60161744582671382560000000000000000000 sqrt(332929 y  + 1000000))
4                                                          2
+ y  (3947525835840000000000000000000000000000 sqrt(0.332929 y  + 1)
2
- 112645908725371144582586400000000000000 sqrt(332929 y  + 1000000))
2
+ 12530880000000000000000000000000000000000 sqrt(0.332929 y  + 1))
4                2
/(- 30948602552985600000000000000000000000000 y  sqrt(0.332929 y  + 1)
2                2
- 185917133116108800000000000000000000000000 y  sqrt(0.332929 y  + 1)
2
- 279214387200000000000000000000000000000000 sqrt(0.332929 y  + 1)) dy
(%i23) ```
Run Example
```alpha_1:1;
(%o1)                                  1
(%i2) Fr_1:2^0.5;
(%o2)                          1.414213562373095
(%i3) x0_1:%i*0.577;
(%o3)                              0.577 %i
(%i4) beta_1:(2*(alpha_1+1)/(3+2*alpha_1));
4
(%o4)                                  -
5
(%i5) A_1:(3+2*alpha_1)*(3+2*alpha_1)*(3+2*alpha_1)/((4*alpha_1+8)*(2*alpha_1+5));
125
(%o5)                                 ---
84
(%i6) C2_1:-0.403;
(%o6)                               - 0.403
(%i7) H1p_1(x):=(125.0/1512)/(1-x*x) - 25.0/196;
125.0
-----
1512     25.0
(%o7)                     H1p_1(x) := ------- - ----
1 - x x   196
(%i8) DiffH1p_1(x):= ''(diff(H1p_1(x), x));
0.16534391534392 x
(%o8)                 DiffH1p_1(x) := ------------------
2 2
(1 - x )
(%i9) F1_1(x):=(1-x*x)^(-alpha_1/2)*assoc_legendre_q(8,-1,x);
- alpha_1
---------
2
(%o9)      F1_1(x) := (1 - x x)          assoc_legendre_q(8, - 1, x)
(%i10) DiffF1_1(x):= ''(diff(F1_1(x), x));
x + 1   10          9
(%o10)/R/ DiffF1_1(x) := - (45045 log(- -----) x   - 90090 x
x - 1
x + 1   8           7                x + 1   6           5
- 135135 log(- -----) x  + 240240 x  + 145530 log(- -----) x  - 218988 x
x - 1                                x - 1
x + 1   4          3               x + 1   2
- 66150 log(- -----) x  + 76464 x  + 11025 log(- -----) x  - 7562 x
x - 1                              x - 1
x + 1          4         2
- 315 log(- -----))/(2304 x  - 4608 x  + 2304)
x - 1
(%i11) H1_1(x)=C2_1*F1_1(x) +H1p_1(x);

rat: replaced -0.403 by -403/1000 = -0.403

rat: replaced -0.1275510204082 by -25/196 = -0.1275510204082

rat: replaced 0.082671957671958 by 125/1512 = 0.082671957671958
x + 1   9               8
(%o11)/R/ H1_1(x) = (1906079175 log(- -----) x  - 3812158350 x
x - 1
x + 1   7               6                    x + 1   5
- 4574590020 log(- -----) x  + 7878460590 x  + 3694861170 log(- -----) x
x - 1                                        x - 1
4                    x + 1   3              2
- 5102427330 x  - 1119654900 log(- -----) x  + 845310978 x
x - 1
x + 1                             2
+ 93304575 log(- -----) x + 41335552)/(1693440000 x  - 1693440000)
x - 1
(%i12) DiffH1_1(x):= DiffF1_1(x)*C2_1+DiffH1p_1(x);
(%o12)          DiffH1_1(x) := DiffF1_1(x) C2_1 + DiffH1p_1(x)
(%i13) H0_1(x):=1.0/(Fr_1*Fr_1) - 1.0/(4*alpha_1+4) + x^2/(4*alpha_1+4);
2
1.0           1.0             x
(%o13)       H0_1(x) := --------- - ------------- + -------------
Fr_1 Fr_1   4 alpha_1 + 4   4 alpha_1 + 4
(%i14) DiffH0_1(x):= ''(diff(H0_1(x), x));
x
(%o14)                         DiffH0_1(x) := -
4
(%i15) B_1:(3+2*alpha_1)*(3+2*alpha_1)*(3+2*alpha_1)/((8*alpha_1+20)*(alpha_1+1));
125
(%o15)                                ---
56
(%i16) Fr_2:2^0.5;
(%o16)                         1.414213562373095
(%i17) U1_Al1(zeta):=realpart( (-beta_1/2)*( DiffH1_1(zeta*x0_1)*x0_1 -A_1*DiffH0_1(zeta)/H0_1(zeta)));
- beta_1
(%o17) U1_Al1(zeta) := realpart(--------
2
A_1 DiffH0_1(zeta)
(DiffH1_1(zeta x0_1) x0_1 - ------------------))
H0_1(zeta)
(%i18) Phi1_Al1(y):= B_1/H0_1(y);
B_1
(%o18)                      Phi1_Al1(y) := -------
H0_1(y)
(%o19)        /usr/share/maxima/5.21.1/share/numeric/interpol.mac
(%i20) p:[[0.000,-5.58],[0.101,-5.57],[0.202,-5.55],[0.303,-5.50],[0.404,-5.41],[0.505,-5.29],[0.606,-5.14],[0.707,-4.97],[0.808,-4.77],[0.909,-4.53]];
(%o20) [[0.0, - 5.58], [0.101, - 5.57], [0.202, - 5.55], [0.303, - 5.5],
[0.404, - 5.41], [0.505, - 5.29], [0.606, - 5.14], [0.707, - 4.97],
[0.808, - 4.77], [0.909, - 4.53]]
(%i21) p1:[[0.000,-5],[0.101,-4.72],[0.202,-4.70],[0.303,-4.67],[0.404,-4.61],[0.505,-4.52],[0.606,-4.41],[0.707,-4.27],[0.808,-4.10],[0.909,-3.90]];
(%o21) [[0.0, - 5], [0.101, - 4.72], [0.202, - 4.7], [0.303, - 4.67],
[0.404, - 4.61], [0.505, - 4.52], [0.606, - 4.41], [0.707, - 4.27],
[0.808, - 4.1], [0.909, - 3.9]]
(%i22) p2:[[0.000,-1.67],[0.101,-1.68],[0.202,-1.67],[0.303,-1.67],[0.404,-1.67],[0.505,-1.67],[0.606,-1.66],[0.707,-1.65],[0.808,-1.63],[0.909,-1.58]];
(%o22) [[0.0, - 1.67], [0.101, - 1.68], [0.202, - 1.67], [0.303, - 1.67],
[0.404, - 1.67], [0.505, - 1.67], [0.606, - 1.66], [0.707, - 1.65],
[0.808, - 1.63], [0.909, - 1.58]]
(%i23) f(x):=cspline(p);
(%o23)                        f(x) := cspline(p)
(%i24) g(x):=cspline(p1);
(%o24)                        g(x) := cspline(p1)
(%i25) h(x):=cspline(p2);
(%o25)                        h(x) := cspline(p2)
(%i26) plot2d([-f(x),-g(x),-h(x), Phi1_Al1(x)],[x,0,1],[xlabel, "y"],[ylabel, "-Phi1(y)"],[style, [lines,2,4],[lines,2,3], [lines,2,2],[lines,5,5]] );
plot```

### Related Help

Help for Assoc_legendre_q