SandorSzabo

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These are replies submitted by SandorSzabo

Yes. A year ago I heard about identify in this forum first (it is a very good forum, I learn a lot from your answers).

The function has removable singularity at  0,  because  the integrand is  [;-\frac{5}{12}+\frac{1}{4}x-\frac{61}{720}x^2+O(x^3);].

I tried to use Mellin transformation and let s->1, but no result. I'm thinking about to transform the original integral to

a complex variable function and consider the original integral as a line integral, but there is no progress.

Sandor

 

I don't remember exactly, but a

select(expression)

     case('A') .....

     case('B').....

     other .....

type of choosing would be useful. Sorry if this description is wrong, hopefully experienced

programmers will guess what I would like.

Sorry, probably I was not clear enough. In fact, my question was not "how to solve this exercise", but "does exist in Maple somewhere in a package where I can define the region Cylindrical which I could use in integration. In VectorCalculus there is SetCoordinates('cylindrical'[r, theta,z]); but I don't know it can be applied and how, or not. Thanks, Sandor
Sorry, probably I was not clear enough. In fact, my question was not "how to solve this exercise", but "does exist in Maple somewhere in a package where I can define the region Cylindrical which I could use in integration. In VectorCalculus there is SetCoordinates('cylindrical'[r, theta,z]); but I don't know it can be applied and how, or not. Thanks, Sandor
Yes, absolutely. The missing preallocation also. Probably one could speed up the proc, but others made better procs.
It was a homework for my students. See http://www.math.bme.hu/~sszabo/NumerikusSzimbolikus/LinearisAlgebra.html and http://www.math.bme.hu/~sszabo/NumerikusSzimbolikus/MatrixTenzorMultiply.pdf and the Maple9 worksheet http://www.math.bme.hu/~sszabo/NumerikusSzimbolikus/MatrixTenzorMultiply.mws There are only 2 Hungarian words (sorry ;-) ) in the worksheet, sor = row, oszlop = column
Your integral is not definite, but indefinite, a class of primitive function, so the actual integral depends on the interval where you want to determine the primitive function. The situation is similar to the multivalued log function in complex function theory or in ordinary diff eq theory the answer may depend on the interval where you want to solve the eq. I made a little simplification
         (sin(x) - 1) sqrt(1 + sin(x))     sqrt(1 - sin(x)) |cos(x)|
         ----------------------------- = - ------------------------- .
                    cos(x)                          cos(x)          
Many thanks for both of you. In the future I will be more deliberate. Sandor
Many thanks for both of you. In the future I will be more deliberate. Sandor
f'(x_0)=0 is not enough. If f' change its sign in x_0, then f has extrema in x_0. Or, a sufficient condition is f''(x_0) not equal to 0. Sandor
f'(x_0)=0 is not enough. If f' change its sign in x_0, then f has extrema in x_0. Or, a sufficient condition is f''(x_0) not equal to 0. Sandor
A shorthand version is a:='a': Sandor
A shorthand version is a:='a': Sandor
Thanks. Sorry for me, I forgot to say, I'm interested in principal value also, if exists. if a = -2
assume(0 less than epsi,epsi less than 0.25);
Int(1/((1-2*y)*sqrt(1-y^2)),y=0..1/2-epsi)=int(1/((1-2*y)*sqrt(1-y^2)),y=0..1/2-epsi);

     /1/2 - epsi                                                         
    |                       1                   1  (1/2)   /   (1/2)    \
    |            ----------------------- dy = - - 3      ln\2 3      + 3/
    |                              (1/2)        6                        
   /0                      /     2\                                      
                 (1 - 2 y) \1 - y /                                      

        1  (1/2)   /   (1/2)    \
      + - 3      ln\2 3      - 3/
        6                        

                   /       /                   (1/2)      \\
        1  (1/2)   |       |     (3 + 2 epsi) 3           ||
      + - 3      Re|arctanh|------------------------------||
        3          |       |                         (1/2)||
                   |       |  /       2             \     ||
                   \       \3 \-4 epsi  + 3 + 4 epsi/     //
Int(1/((1-2*y)*sqrt(1-y^2)),y=1/2+epsi..1)=int(1/((1-2*y)*sqrt(1-y^2)),y=1/2+epsi..1);

          /1                                                /       /
         |                     1                 1  (1/2)   |       |
         |          ----------------------- dy = - 3      Re|arctanh|
         |                            (1/2)      3          |       |
        /1/2 + epsi           /     2\                      |       |
                    (1 - 2 y) \1 - y /                      \       \

                              (1/2)     \\
               (-3 + 2 epsi) 3          ||
          ------------------------------||
                                   (1/2)||
            /       2             \     ||
          3 \-4 epsi  + 3 - 4 epsi/     //
with(MultiSeries):

Arctanh( 1/3*(-3+2/Delta)*sqrt(3)/sqrt(-4/Delta^2+3-4/Delta))=asympt( arctanh( 1/3*(-3+2/Delta)*sqrt(3)/sqrt(-4/Delta^2+3-4/Delta)), Delta,2);

         /     /       2  \  (1/2)     \                                  
         |     |-3 + -----| 3          |                                  
         |     \     Delta/            |     1                     1   /8\
  Arctanh|-----------------------------| = - - ln(2) - ln(Delta) + - ln|-|
         |                        (1/2)|     2                     2   \9/
         |  /    4            4  \     |                                  
         |3 |- ------ + 3 - -----|     |                                  
         |  |       2       Delta|     |                                  
         \  \  Delta             /     /                                  

       1         /  1  \
     + - I Pi + O|-----|
       2         \Delta/
Arctanh( 1/3*(3+2/Delta)*sqrt(3)/sqrt(-4/Delta^2+3+4/Delta))=asympt( arctanh(1/3*(3+2/Delta)*sqrt(3)/sqrt(-4/Delta^2+3+4/Delta)),Delta,2);

          /     /      2  \  (1/2)      \                                
          |     |3 + -----| 3           |                                
          |     \    Delta/             |   1                     1   /8\
   Arctanh|-----------------------------| = - ln(2) + ln(Delta) - - ln|-|
          |                        (1/2)|   2                     2   \9/
          |  /    4            4  \     |                                
          |3 |- ------ + 3 + -----|     |                                
          |  |       2       Delta|     |                                
          \  \  Delta             /     /                                

        1         /  1  \
      - - I Pi + O|-----|
        2         \Delta/
-1/6*sqrt(3)*ln(2*sqrt(3)+3)+1/6*sqrt(3)*ln(2*sqrt(3)-3)+1/3*sqrt(3)*(-1/2*ln(2)-ln(Delta)+1/2*ln(8/9)+1/2*I*Pi+1/2*ln(2)+ln(Delta)-1/2*ln(8/9)-1/2*I*Pi);

             1  (1/2)   /   (1/2)    \   1  (1/2)   /   (1/2)    \
           - - 3      ln\2 3      + 3/ + - 3      ln\2 3      - 3/
             6                           6                        

If a is not equal to -2 but less than -1
assume(a less than -1);
Int(1/((1+a*y)*sqrt(1-y^2)),y=1/2+epsi..1)=int(1/((1+a*y)*(sqrt(1-y^2))),y=1/2+epsi..1);

  /1                                       
 |                     1                   
 |          ----------------------- dy = - 
 |                            (1/2)        
/1/2 + epsi           /     2\             
            (1 + a y) \1 - y /             

     /     /       /           a + y           \      1              \\       
  -2 |limit|arctanh|---------------------------|, y = - + epsi, right|| + I Pi
     |     |       |        (1/2)         (1/2)|      2              ||       
     |     |       |/ 2    \      /     2\     |                     ||       
     \     \       \\a  - 1/      \1 - y /     /                     //       
  ----------------------------------------------------------------------------
                                          (1/2)                               
                                  / 2    \                                    
                                2 \a  - 1/                                    
Int(1/((1+a*y)*sqrt(1-y^2)),y=0..1/2-epsi)=int(1/((1+a*y)*sqrt(1-y^2)),y=0..1/2-epsi);

    /1/2 - epsi                                                   
   |                       1                        1        /    
   |            ----------------------- dy = --------------- |I Pi
   |                              (1/2)                (1/2) |    
  /0                      /     2\             / 2    \      |    
                (1 + a y) \1 - y /           2 \a  - 1/      \    

           /       /      a      \\
     + 2 Re|arctanh|-------------||
           |       |        (1/2)||
           |       |/ 2    \     ||
           \       \\a  - 1/     //

         /     /       /           a + y           \      1             \\\
     - 2 |limit|arctanh|---------------------------|, y = - - epsi, left|||
         |     |       |        (1/2)         (1/2)|      2             |||
         |     |       |/ 2    \      /     2\     |                    |||
         \     \       \\a  - 1/      \1 - y /     /                    ///
solve( (A+1/2)/(sqrt(A^2-1)*sqrt(3/4))=-1,A);

                                     -2
Arctanh((a+1/2+1/Delta)/(sqrt(a^2-1)*sqrt(1-(1/2+1/Delta)^2)))=asympt( arctanh((a+1/2+1/Delta)/(sqrt(a^2-1)*sqrt(1-(1/2+1/Delta)^2))), Delta,2);

          /                1     1              \           
          |            a + - + -----            |          /
          |                2   Delta            |          |
   Arctanh|-------------------------------------| = arctanh|
          |                                (1/2)|          |
          |        (1/2) /               2\     |          |
          |/ 2    \      |    /1     1  \ |     |          \
          |\a  - 1/      |1 - |- + -----| |     |           
          \              \    \2   Delta/ /     /           

                                          (1/2)                          
      (1/2)  (1/2)          \     / 2    \       (1/2)  (1/2)            
     3      4      (2 a + 1)|   2 \a  - 1/      3      4         /  1   \
     -----------------------| - ----------------------------- + O|------|
                   (1/2)    |          3 (a + 2) Delta           |     2|
           / 2    \         |                                    \Delta /
         6 \a  - 1/         /                                            


Thanks. Sorry for me, I forgot to say, I'm interested in principal value also, if exists. if a = -2
assume(0 less than epsi,epsi less than 0.25);
Int(1/((1-2*y)*sqrt(1-y^2)),y=0..1/2-epsi)=int(1/((1-2*y)*sqrt(1-y^2)),y=0..1/2-epsi);

     /1/2 - epsi                                                         
    |                       1                   1  (1/2)   /   (1/2)    \
    |            ----------------------- dy = - - 3      ln\2 3      + 3/
    |                              (1/2)        6                        
   /0                      /     2\                                      
                 (1 - 2 y) \1 - y /                                      

        1  (1/2)   /   (1/2)    \
      + - 3      ln\2 3      - 3/
        6                        

                   /       /                   (1/2)      \\
        1  (1/2)   |       |     (3 + 2 epsi) 3           ||
      + - 3      Re|arctanh|------------------------------||
        3          |       |                         (1/2)||
                   |       |  /       2             \     ||
                   \       \3 \-4 epsi  + 3 + 4 epsi/     //
Int(1/((1-2*y)*sqrt(1-y^2)),y=1/2+epsi..1)=int(1/((1-2*y)*sqrt(1-y^2)),y=1/2+epsi..1);

          /1                                                /       /
         |                     1                 1  (1/2)   |       |
         |          ----------------------- dy = - 3      Re|arctanh|
         |                            (1/2)      3          |       |
        /1/2 + epsi           /     2\                      |       |
                    (1 - 2 y) \1 - y /                      \       \

                              (1/2)     \\
               (-3 + 2 epsi) 3          ||
          ------------------------------||
                                   (1/2)||
            /       2             \     ||
          3 \-4 epsi  + 3 - 4 epsi/     //
with(MultiSeries):

Arctanh( 1/3*(-3+2/Delta)*sqrt(3)/sqrt(-4/Delta^2+3-4/Delta))=asympt( arctanh( 1/3*(-3+2/Delta)*sqrt(3)/sqrt(-4/Delta^2+3-4/Delta)), Delta,2);

         /     /       2  \  (1/2)     \                                  
         |     |-3 + -----| 3          |                                  
         |     \     Delta/            |     1                     1   /8\
  Arctanh|-----------------------------| = - - ln(2) - ln(Delta) + - ln|-|
         |                        (1/2)|     2                     2   \9/
         |  /    4            4  \     |                                  
         |3 |- ------ + 3 - -----|     |                                  
         |  |       2       Delta|     |                                  
         \  \  Delta             /     /                                  

       1         /  1  \
     + - I Pi + O|-----|
       2         \Delta/
Arctanh( 1/3*(3+2/Delta)*sqrt(3)/sqrt(-4/Delta^2+3+4/Delta))=asympt( arctanh(1/3*(3+2/Delta)*sqrt(3)/sqrt(-4/Delta^2+3+4/Delta)),Delta,2);

          /     /      2  \  (1/2)      \                                
          |     |3 + -----| 3           |                                
          |     \    Delta/             |   1                     1   /8\
   Arctanh|-----------------------------| = - ln(2) + ln(Delta) - - ln|-|
          |                        (1/2)|   2                     2   \9/
          |  /    4            4  \     |                                
          |3 |- ------ + 3 + -----|     |                                
          |  |       2       Delta|     |                                
          \  \  Delta             /     /                                

        1         /  1  \
      - - I Pi + O|-----|
        2         \Delta/
-1/6*sqrt(3)*ln(2*sqrt(3)+3)+1/6*sqrt(3)*ln(2*sqrt(3)-3)+1/3*sqrt(3)*(-1/2*ln(2)-ln(Delta)+1/2*ln(8/9)+1/2*I*Pi+1/2*ln(2)+ln(Delta)-1/2*ln(8/9)-1/2*I*Pi);

             1  (1/2)   /   (1/2)    \   1  (1/2)   /   (1/2)    \
           - - 3      ln\2 3      + 3/ + - 3      ln\2 3      - 3/
             6                           6                        

If a is not equal to -2 but less than -1
assume(a less than -1);
Int(1/((1+a*y)*sqrt(1-y^2)),y=1/2+epsi..1)=int(1/((1+a*y)*(sqrt(1-y^2))),y=1/2+epsi..1);

  /1                                       
 |                     1                   
 |          ----------------------- dy = - 
 |                            (1/2)        
/1/2 + epsi           /     2\             
            (1 + a y) \1 - y /             

     /     /       /           a + y           \      1              \\       
  -2 |limit|arctanh|---------------------------|, y = - + epsi, right|| + I Pi
     |     |       |        (1/2)         (1/2)|      2              ||       
     |     |       |/ 2    \      /     2\     |                     ||       
     \     \       \\a  - 1/      \1 - y /     /                     //       
  ----------------------------------------------------------------------------
                                          (1/2)                               
                                  / 2    \                                    
                                2 \a  - 1/                                    
Int(1/((1+a*y)*sqrt(1-y^2)),y=0..1/2-epsi)=int(1/((1+a*y)*sqrt(1-y^2)),y=0..1/2-epsi);

    /1/2 - epsi                                                   
   |                       1                        1        /    
   |            ----------------------- dy = --------------- |I Pi
   |                              (1/2)                (1/2) |    
  /0                      /     2\             / 2    \      |    
                (1 + a y) \1 - y /           2 \a  - 1/      \    

           /       /      a      \\
     + 2 Re|arctanh|-------------||
           |       |        (1/2)||
           |       |/ 2    \     ||
           \       \\a  - 1/     //

         /     /       /           a + y           \      1             \\\
     - 2 |limit|arctanh|---------------------------|, y = - - epsi, left|||
         |     |       |        (1/2)         (1/2)|      2             |||
         |     |       |/ 2    \      /     2\     |                    |||
         \     \       \\a  - 1/      \1 - y /     /                    ///
solve( (A+1/2)/(sqrt(A^2-1)*sqrt(3/4))=-1,A);

                                     -2
Arctanh((a+1/2+1/Delta)/(sqrt(a^2-1)*sqrt(1-(1/2+1/Delta)^2)))=asympt( arctanh((a+1/2+1/Delta)/(sqrt(a^2-1)*sqrt(1-(1/2+1/Delta)^2))), Delta,2);

          /                1     1              \           
          |            a + - + -----            |          /
          |                2   Delta            |          |
   Arctanh|-------------------------------------| = arctanh|
          |                                (1/2)|          |
          |        (1/2) /               2\     |          |
          |/ 2    \      |    /1     1  \ |     |          \
          |\a  - 1/      |1 - |- + -----| |     |           
          \              \    \2   Delta/ /     /           

                                          (1/2)                          
      (1/2)  (1/2)          \     / 2    \       (1/2)  (1/2)            
     3      4      (2 a + 1)|   2 \a  - 1/      3      4         /  1   \
     -----------------------| - ----------------------------- + O|------|
                   (1/2)    |          3 (a + 2) Delta           |     2|
           / 2    \         |                                    \Delta /
         6 \a  - 1/         /                                            


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