The ODE on wich I am working is

> deq;

                      /  2      \                                
                      | d       |           / d      \           
            x (1 - x) |---- y(x)| + (1 - x) |--- y(x)| - y(x) = 0
                      |   2     |           \ dx     /           
                      \ dx      /                                

|x|<1

Maple give

>y1:=rhs( dsolve(deq, y(x)));

                          /                            1   \         I
      y1:=   _C1 hypergeom|[-I, 1 - I], [1 - 2 I], - ------| (-1 + x) 
                          \                          -1 + x/          

                        /                           1   \         (-I)
         + _C2 hypergeom|[I, 1 + I], [1 + 2 I], - ------| (-1 + x)    
                        \                         -1 + x/             

So if I understand, the only way to go is

> f :=  (s,x) -> 1+sum(GAMMA(n-I)*GAMMA(n+I)*x^n/(GAMMA(-I)*GAMMA(I)*factorial(n)^2), n = 1 .. s) ;

                        /   s                                    \
                        | -----                                  |
                        |  \                                  n  |
                        |   )      GAMMA(n - I) GAMMA(n + I) x   |
        f:= (s,x)-> 1 + |  /     --------------------------------|
                        | -----                                 2|
                        \ n = 1  GAMMA(-I) GAMMA(I) factorial(n) /

> plot(Re@evalf[50](f(100,x))), x = 0 .. 1);

Warning, unable to ............

Maybe I don't understand correctly!

More input please Acer..

and I have with assistant,special function the sum-form of the Psi(x)  fonction

Thanks

I know that it is a straight line.  This was just a simple example but yes I was expecting a NULL return.  For a more complex equation, it seems, if I understand correctly, that I will have to solve than check if there are roots that are in the range I am interested.  Or simply take advantage of the graphing possibilities of Maple.

Thanks again Acer and Georgios

 and if I am still stuck, I'll get back to you.  Thanks for the trouble.

 the second, FormalPowerSeries, give a result that have no sense at all.  What about the two summations sign.  And there is no roots for _Z.  Do you have at hand an example that yould give the summand of a serie.

But anyway, you solve my problem.  Thanks Robert!

I found that my solution is:

> y1 := x^(1/2)*(sum(factorial(2*n)*(-(1/4)*x)^n/factorial(n)^3, n = 0 .. infinity));

                               (1/2)        /   1  \
                              x      BesselI|0, - x|
                                            \   2  /
                        y1 := ----------------------
                                        /1  \       
                                     exp|- x|       
                                        \2  /       
> diff(y1, x);

             /   1  \     (1/2)        /   1  \    (1/2)        /   1  \
      BesselI|0, - x|    x      BesselI|1, - x|   x      BesselI|0, - x|
             \   2  /                  \   2  /                 \   2  /
     ----------------- + ---------------------- - ----------------------
        (1/2)    /1  \              /1  \                    /1  \      
     2 x      exp|- x|         2 exp|- x|               2 exp|- x|      
                 \2  /              \2  /                    \2  /      

I[0] (that's what is appearing on my screen) is the modified Bessel function of the first kind of zero order.  So I assume that I[1] is the modified Bessel function of the first kind of order one.

Is there a link between them.  Sorry to ask here cause I don't have any books on specials functions.

Thanks in advance

 All I  need to do is digest how it is working, since it's exactly what I wanted.

 op(1,op(1,e1))

or

> op(1, e1[1]);

 

 I suggest to go in the help menu of Maple and choose "Take a tour of Maple".  It will give you a good head start.

 I forgot to write n:=i+2;

 HI,

> F := x ->   sin(x^2)^3+cos(x^2)^2;

                                         3          2
                                     / 2\       / 2\ 
                        F := x -> sin\x /  + cos\x / 
> H := t -> 1/tan(exp(t));

                                           1     
                            H := t -> -----------
                                      tan(exp(t))
> Z := F(H(t));

                                      3                    2
                        /     1      \       /     1      \ 
                Z := sin|------------|  + cos|------------| 
                        |           2|       |           2| 
                        \tan(exp(t)) /       \tan(exp(t)) / 
> g := unapply(Z, t);

                                         3                    2
                           /     1      \       /     1      \ 
              g := t -> sin|------------|  + cos|------------| 
                           |           2|       |           2| 
                           \tan(exp(t)) /       \tan(exp(t)) / 
> evalf(g(1));

                                -0.8881757633
So everything seems to work fine!
> dg := (D(g))(sqrt(2));

dg := 
                    /                        2                        / 
          1         |     /        1        \     /        1        \ | 
- ----------------- |6 sin|-----------------|  cos|-----------------| \1
                  3 |     |                2|     |                2|   
     /   / (1/2)\\  |     |   /   / (1/2)\\ |     |   /   / (1/2)\\ |   
  tan\exp\2     //  \     \tan\exp\2     // /     \tan\exp\2     // /   

                     2\            \                     /      
        /   / (1/2)\\ |    / (1/2)\|           1         |     /
   + tan\exp\2     // / exp\2     /| + ----------------- |4 cos|
                                   |                   3 |     |
                                   |      /   / (1/2)\\  |     |
                                   /   tan\exp\2     //  \     \

                                            /                    2\            
          1        \    /        1        \ |       /   / (1/2)\\ |    / (1/2)\
  -----------------| sin|-----------------| \1 + tan\exp\2     // / exp\2     /
                  2|    |                2|                                    
     /   / (1/2)\\ |    |   /   / (1/2)\\ |                                    
  tan\exp\2     // /    \tan\exp\2     // /                                    

  \
  |
  |
  |
  |
  /
> evalf(%);

                                 2.154213500
So for the moment, dg is a number
> dg := unapply((D(g))(t), t);

                                2                                            
                  /     1      \     /     1      \ /               2\       
             6 sin|------------|  cos|------------| \1 + tan(exp(t)) / exp(t)
                  |           2|     |           2|                          
                  \tan(exp(t)) /     \tan(exp(t)) /                          
dg := t -> - ----------------------------------------------------------------
                                                  3                          
                                       tan(exp(t))                           

          /     1      \    /     1      \ /               2\       
     4 cos|------------| sin|------------| \1 + tan(exp(t)) / exp(t)
          |           2|    |           2|                          
          \tan(exp(t)) /    \tan(exp(t)) /                          
   + ---------------------------------------------------------------
                                         3                          
                              tan(exp(t))                           
> evalf(dg(sqrt(2)));

                                 2.154213500
Now we are ready to evaluate the limit
> limit(dg(t), t = infinity);

                                  undefined
> plot(dg(t), t = 0 .. infinity);


See the result and you will know why the result is undefined

 HI,

First of all, Joe is right.  e is a variable.  While exp(x) is the exponential function.  If you want to use "e" for the exponential, take it from the "Common Symbols" palette.

As for the euler number, just type ?Euler and you will get all the informations you need.