An integral that does not give what I am expecting

Posted on 2008-11-22 13:12 By lemelinm ( Maple Rating 3

235)

Categories:

How do I ...? with Maple

Hi all,

> eq1 := Int(f(k, x)^2, x);

                                  /         2       
                                 |      / k\        
                                 |      \x /        
                         eq1 :=  |  ------------- dx
                                 |              2   
                                 |  /     (2 k)\    
                                /   \1 + x     /    

> eq2 := Int(x^(2*k)/(1+x^(2*k))^2, x = 0 .. 1);

                                 /1                 
                                |       (2 k)       
                                |      x            
                        eq2 :=  |   ------------- dx
                                |               2   
                               /0   /     (2 k)\    
                                    \1 + x     /    
> with(IntegrationTools);
> eq3 := `assuming`([Change(eq2, u = x^(2*k), u)], [k::posint]);

                                    /1   / 1 \    
                                   |     |---|    
                                   |     \2 k/    
                                   |    u         
                                   |   -------- du
                                   |          2   
                                  /0   (1 + u)    
                           eq3 := ----------------
                                        2 k       
> `assuming`([value(eq3)], [k::posint]);

                            /[   2 k + 1]  [1 + 4 k]    \
                   hypergeom|[2, -------], [-------], -1|
                            \[     2 k  ]  [  2 k  ]    /
                   --------------------------------------
                                  2 k + 1                
> `assuming`([limit(%, k = infinity)], [k::posint]);

              /         /[   2 k + 1]  [1 + 4 k]    \              \
              |hypergeom|[2, -------], [-------], -1|              |
              |         \[     2 k  ]  [  2 k  ]    /              |
         limit|--------------------------------------, k = infinity|
              \               2 k + 1                              /

I am expecting the limit to be 0. In Maple 10, I have

> with(student)

>                 /[         /      (2 k)       \]         \

>       `assuming`\[changevar\eq1, x      = u, u/], [0 < k]/

Error, (in assuming) when calling '`student/powsubs`'. Received: 'usage: powsubs(a=b , f ) '

I have the same problem in Maple 6.  What am I doing wrong?

Mario

partial solution

Comment on 2008-11-22 14:10 from Joe Riel ( Maple Rating 4

1260)

While it may not solve your problem, you can compute the asymptote of the transformed integral:

evalindets(eq3, Int(anything$2), limit, k=infinity);
                                      1
                                      ---
                                      4 k

MeijerG

Comment on 2008-11-22 14:46 from acer ( Maple Rating 4

1410)

> eq2 := Int(x^(2*k)/(1+x^(2*k))^2, x = 0 .. 1):
> with(IntegrationTools):
> eq3 := Change(eq2, u = x^(2*k), u) assuming k::posint:

> value(eq3) assuming k::posint;

                                  2 k + 1    4 k + 1
                    hypergeom([2, -------], [-------], -1)
                                    2 k        2 k
                    --------------------------------------
                                   2 k + 1
 
> limit(convert(%,MeijerG), k = infinity) assuming k::posint;
                                       0

> limit(convert(%%,MeijerG), k = infinity);
                                       0

acer

other, simple solution

Comment on 2008-11-22 14:52 from Axel Vogt ( Maple Rating 4

994)

What you do is to write Euler's integral as 2F1, but it is not
continuous in its parameters (else it would be 1/2 in that case)
and Maple does not do it.

To prove it: 

  eq3; combine(%);
  op(1,%);
  limit(%,k=infinity);
                                  0

So the integrand is 0 in the limit. Now justify interchanging
limit and integral to get your desired result.

An integral that does not give what I am expecting

partial solution

MeijerG

other, simple solution

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