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SandorSzabo

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Complex function...

SandorSzabo 607
March 05 2008

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

 

select case...

SandorSzabo 607
February 05 2008

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.

Coordinate System...

SandorSzabo 607
January 16 2008
0 0
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

Coordinate System...

SandorSzabo 607
January 16 2008
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...

SandorSzabo 607
December 14 2007
Yes, absolutely. The missing preallocation also. Probably one could speed up the proc, but others made better procs.

KroneckerProduct...

SandorSzabo 607
December 01 2007
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

Definite Integration...

SandorSzabo 607
November 28 2007
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)

Division...

SandorSzabo 607
November 28 2007
0 0
Many thanks for both of you. In the future I will be more deliberate. Sandor

Division...

SandorSzabo 607
November 28 2007
Many thanks for both of you. In the future I will be more deliberate. Sandor

Necessary and sufficient conds for extre...

SandorSzabo 607
November 16 2007
0 0
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

Necessary and sufficient conds for extre...

SandorSzabo 607
November 16 2007
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

unassume...

SandorSzabo 607
October 05 2007
0 0
A shorthand version is a:='a': Sandor

unassume...

SandorSzabo 607
October 05 2007
A shorthand version is a:='a': Sandor

Cauchy principal value...

SandorSzabo 607
September 22 2007
0 0
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/         /

Cauchy principal value...

SandorSzabo 607
September 22 2007
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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