Singularities of Møller-Plesset series: example "a8"

[n₁, n₂, n₃] approximant is defined as a solution of the quadratic equation
A(z)f² + B(z)f + C(z) = 0
with polynomial coefficients A(z), B(z) and C(z) of degree n₃, n₂ and n₁ respectively.

Square-root singularities are determined as zeroes of the discriminant
D(z) = B²(z) - 4A(z)C(z).
The weight c of the singularity z_c is defined so that
f ~ c(1 - z/z_c)^1/2 at z -> z_c.
The weight is calculated by formula
c = 1/2[-z(D/A²)']^1/2
where r. h. s. of the above equation is evaluated at z = z_c.

Table 1. Singularities with their weights for the quadratic approximant [1, 0, 0]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.1098

0.163
Singularities of quadratic [1, 0, 0] approximant

  Top of the page

Table 2. Singularities with their weights for the quadratic approximant [1, 1, 0]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.4066

0.268

2

88.8986

2.13 i

  Top of the page

Table 3. Singularities with their weights for the quadratic approximant [1, 1, 1]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.5231

0.379
Singularities of quadratic [1, 1, 1] approximant

2

-23.1777

0.731

  Top of the page

Table 4. Singularities with their weights for the quadratic approximant [2, 1, 1]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.2212

0.11
Singularities of quadratic [2, 1, 1] approximant

2

1.9046

0.204 i

3

3.3308

0.699

  Top of the page

Table 5. Singularities with their weights for the quadratic approximant [2, 2, 1]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.4404

0.273
Singularities of quadratic [2, 2, 1] approximant

2

-7.4525 + 1.4664 i

0.66 + 0.323 i

3

-7.4525 - 1.4664 i

0.66 - 0.323 i

4

28.4954

2.07 i

  Top of the page

Table 6. Singularities with their weights for the quadratic approximant [2, 2, 2]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.6843

4.37
Singularities of quadratic [2, 2, 2] approximant

2

2.0332

1.29 i

3

3.3141

24.7

4

-7.0871

0.215

  Top of the page

Table 7. Singularities with their weights for the quadratic approximant [3, 2, 2]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.595 + 0.3265 i

0.309 - 0.222 i
Singularities of quadratic [3, 2, 2] approximant

2

1.595 - 0.3265 i

0.309 + 0.222 i

3

2.2951

0.641

4

-5.7901

0.156

5

-20.7909

0.186 i

  Top of the page

Table 8. Singularities with their weights for the quadratic approximant [3, 3, 2]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.5733 + 0.3243 i

0.269 - 0.237 i
Singularities of quadratic [3, 3, 2] approximant

2

1.5733 - 0.3243 i

0.269 + 0.237 i

3

2.1857

0.493

4

-6.0635

0.181

5

-23.5914

0.215 i

6

1245.2983

1.93 i

  Top of the page

Table 9. Singularities with their weights for the quadratic approximant [3, 3, 3]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.5729 + 0.3239 i

0.268 - 0.239 i
Singularities of quadratic [3, 3, 3] approximant

2

1.5729 - 0.3239 i

0.268 + 0.239 i

3

2.1827

0.489

4

-6.0709

0.184

5

-21.9614

0.224 i

6

-418.9354

2.16

  Top of the page

Table 10. Singularities with their weights for the quadratic approximant [4, 3, 3]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

0.5746 + 0.e-4 i

0.000897 - 0.000897 i
Singularities of quadratic [4, 3, 3] approximant

2

0.5746 - 0.e-4 i

0.000897 + 0.000897 i

3

1.5505 + 0.2966 i

0.128 - 0.332 i

4

1.5505 - 0.2966 i

0.128 + 0.332 i

5

2.1218

0.39

6

-5.8007

0.14

7

-28.5096

0.192 i

  Top of the page

Table 11. Singularities with their weights for the quadratic approximant [4, 4, 3]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.5902 + 0.366 i

0.272 - 0.0715 i
Singularities of quadratic [4, 4, 3] approximant

2

1.5902 - 0.366 i

0.272 + 0.0715 i

3

2.346

0.97

4

-3.8027

0.026

5

-4.9206

0.0275 i

6

-4.6869 + 7.0614 i

0.0772 - 0.0877 i

7

-4.6869 - 7.0614 i

0.0772 + 0.0877 i

8

33.7868

5.52 i

  Top of the page

Table 12. Singularities with their weights for the quadratic approximant [4, 4, 4]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.5912 + 0.3735 i

0.27 - 0.029 i
Singularities of quadratic [4, 4, 4] approximant

2

1.5912 - 0.3735 i

0.27 + 0.029 i

3

2.248

0.772

4

-0.6592 + 2.4583 i

0.00443 + 0.0109 i

5

-0.6592 - 2.4583 i

0.00443 - 0.0109 i

6

-0.679 + 2.5177 i

0.0113 - 0.00419 i

7

-0.679 - 2.5177 i

0.0113 + 0.00419 i

8

-4.9634

0.0503

  Top of the page

Table 13. Singularities with their weights for the quadratic approximant [5, 4, 4]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.5787 + 0.2702 i

0.0797 + 0.372 i
Singularities of quadratic [5, 4, 4] approximant

2

1.5787 - 0.2702 i

0.0797 - 0.372 i

3

2.2061 + 0.727 i

0.224 + 0.211 i

4

2.2061 - 0.727 i

0.224 - 0.211 i

5

-2.4712 + 0.0271 i

0.00306 + 0.00308 i

6

-2.4712 - 0.0271 i

0.00306 - 0.00308 i

7

3.2013

0.561

8

-4.9032

0.0325

9

16.4278

0.564 i

  Top of the page

Table 14. Singularities with their weights for the quadratic approximant [5, 5, 4]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.3998 + 0.0329 i

0.0394 - 0.0367 i
Singularities of quadratic [5, 5, 4] approximant

2

1.3998 - 0.0329 i

0.0394 + 0.0367 i

3

1.6915 + 0.4481 i

0.0679 + 0.341 i

4

1.6915 - 0.4481 i

0.0679 - 0.341 i

5

2.1807

1.16

6

-3.7952 + 0.0761 i

0.102 + 0.0968 i

7

-3.7952 - 0.0761 i

0.102 - 0.0968 i

8

-9.4334

1.08

9

-17.5377

1.01 i

10

34.1665

1.23 i

  Top of the page

Table 15. Singularities with their weights for the quadratic approximant [5, 5, 5]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.4097 + 0.0365 i

0.0416 - 0.038 i
Singularities of quadratic [5, 5, 5] approximant

2

1.4097 - 0.0365 i

0.0416 + 0.038 i

3

1.6983 + 0.4525 i

0.0356 + 0.345 i

4

1.6983 - 0.4525 i

0.0356 - 0.345 i

5

2.1646

1.15

6

-3.8295 + 0.0977 i

0.0787 + 0.0736 i

7

-3.8295 - 0.0977 i

0.0787 - 0.0736 i

8

-10.7667 + 2.519 i

0.434 + 1.18 i

9

-10.7667 - 2.519 i

0.434 - 1.18 i

10

66.1385

0.427 i

  Top of the page

Table 16. Singularities with their weights for the quadratic approximant [6, 5, 5]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.4258 + 0.0499 i

0.038 - 0.0309 i
Singularities of quadratic [6, 5, 5] approximant

2

1.4258 - 0.0499 i

0.038 + 0.0309 i

3

1.7378 + 0.4815 i

0.17 - 0.267 i

4

1.7378 - 0.4815 i

0.17 + 0.267 i

5

1.9955

0.743

6

-3.3634 + 0.2346 i

0.00327 + 0.00418 i

7

-3.3634 - 0.2346 i

0.00327 - 0.00418 i

8

-4.2643

0.00754

9

-1.0869 + 10.6297 i

0.144 + 0.088 i

10

-1.0869 - 10.6297 i

0.144 - 0.088 i

11

-13.2713

0.121 i

  Top of the page

Table 17. Singularities with their weights for the quadratic approximant [6, 6, 5]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.4444 + 0.0692 i

0.039 - 0.0286 i
Singularities of quadratic [6, 6, 5] approximant

2

1.4444 - 0.0692 i

0.039 + 0.0286 i

3

1.7233 + 0.5483 i

0.108 - 0.124 i

4

1.7233 - 0.5483 i

0.108 + 0.124 i

5

1.8216

0.528

6

2.862

0.678 i

7

-3.6866 + 0.2224 i

0.015 + 0.0146 i

8

-3.6866 - 0.2224 i

0.015 - 0.0146 i

9

3.7212

1.25

10

-6.8583

0.086

11

18.0958

2.98 i

12

-27.5978

1.28 i

  Top of the page

Table 18. Singularities with their weights for the quadratic approximant [6, 6, 6]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

0.3727 + 0.e-5 i

1.44e-7 - 1.44e-7 i
Singularities of quadratic [6, 6, 6] approximant

2

0.3727 - 0.e-5 i

1.44e-7 + 1.44e-7 i

3

1.1773 + 0.0065 i

0.000619 - 0.0006 i

4

1.1773 - 0.0065 i

0.000619 + 0.0006 i

5

1.4178

0.0127

6

1.6929 + 0.6798 i

0.0406 - 0.00596 i

7

1.6929 - 0.6798 i

0.0406 + 0.00596 i

8

2.8668

0.496 i

9

-3.44

0.00855

10

-4.4027

0.0123 i

11

4.9751

1.09

12

-14.3294

0.222

  Top of the page

Table 19. Singularities with their weights for the quadratic approximant [7, 6, 6]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.3452 + 0.1314 i

0.0405 + 0.0124 i
Singularities of quadratic [7, 6, 6] approximant

2

1.3452 - 0.1314 i

0.0405 - 0.0124 i

3

1.3586 + 0.107 i

0.0153 - 0.0388 i

4

1.3586 - 0.107 i

0.0153 + 0.0388 i

5

1.6981 + 0.4596 i

0.0258 - 0.341 i

6

1.6981 - 0.4596 i

0.0258 + 0.341 i

7

2.0846

0.786

8

-3.609 + 0.2265 i

0.0104 + 0.0114 i

9

-3.609 - 0.2265 i

0.0104 - 0.0114 i

10

-5.6194

0.0335

11

-0.0007 + 13.8331 i

0.235 + 0.203 i

12

-0.0007 - 13.8331 i

0.235 - 0.203 i

13

-35.1917

1.33 i

  Top of the page

Table 20. Singularities with their weights for the quadratic approximant [7, 7, 6]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

1

1.4624

0.0386
Singularities of quadratic [7, 7, 6] approximant

2

1.4272 + 0.3745 i

0.0121 + 0.00503 i

3

1.4272 - 0.3745 i

0.0121 - 0.00503 i

4

1.4101 + 0.439 i

0.00685 - 0.0103 i

5

1.4101 - 0.439 i

0.00685 + 0.0103 i

6

1.9162 + 0.6441 i

0.0472 + 0.0992 i

7

1.9162 - 0.6441 i

0.0472 - 0.0992 i

8

-3.3868

0.00702

9

-4.5445

0.00859 i

10

-5.7371 + 4.805 i

0.00142 - 0.0249 i

11

-5.7371 - 4.805 i

0.00142 + 0.0249 i

12

-4.0523 + 7.3331 i

0.0313 - 0.0184 i

13

-4.0523 - 7.3331 i

0.0313 + 0.0184 i

14

14.9408

13.8 i

  Top of the page

Table 1. Singularities with their weights for the quadratic approximant [1, 0, 0] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.1098	0.163

Table 2. Singularities with their weights for the quadratic approximant [1, 1, 0] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.4066	0.268
2	88.8986	2.13 i

Table 3. Singularities with their weights for the quadratic approximant [1, 1, 1] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.5231	0.379
2	-23.1777	0.731

Table 4. Singularities with their weights for the quadratic approximant [2, 1, 1] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.2212	0.11
2	1.9046	0.204 i
3	3.3308	0.699

Table 5. Singularities with their weights for the quadratic approximant [2, 2, 1] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.4404	0.273
2	-7.4525 + 1.4664 i	0.66 + 0.323 i
3	-7.4525 - 1.4664 i	0.66 - 0.323 i
4	28.4954	2.07 i

Table 6. Singularities with their weights for the quadratic approximant [2, 2, 2] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.6843	4.37
2	2.0332	1.29 i
3	3.3141	24.7
4	-7.0871	0.215

Table 7. Singularities with their weights for the quadratic approximant [3, 2, 2] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.595 + 0.3265 i	0.309 - 0.222 i
2	1.595 - 0.3265 i	0.309 + 0.222 i
3	2.2951	0.641
4	-5.7901	0.156
5	-20.7909	0.186 i

Table 8. Singularities with their weights for the quadratic approximant [3, 3, 2] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.5733 + 0.3243 i	0.269 - 0.237 i
2	1.5733 - 0.3243 i	0.269 + 0.237 i
3	2.1857	0.493
4	-6.0635	0.181
5	-23.5914	0.215 i
6	1245.2983	1.93 i

Table 9. Singularities with their weights for the quadratic approximant [3, 3, 3] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.5729 + 0.3239 i	0.268 - 0.239 i
2	1.5729 - 0.3239 i	0.268 + 0.239 i
3	2.1827	0.489
4	-6.0709	0.184
5	-21.9614	0.224 i
6	-418.9354	2.16

Table 10. Singularities with their weights for the quadratic approximant [4, 3, 3] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	0.5746 + 0.e-4 i	0.000897 - 0.000897 i
2	0.5746 - 0.e-4 i	0.000897 + 0.000897 i
3	1.5505 + 0.2966 i	0.128 - 0.332 i
4	1.5505 - 0.2966 i	0.128 + 0.332 i
5	2.1218	0.39
6	-5.8007	0.14
7	-28.5096	0.192 i

Table 11. Singularities with their weights for the quadratic approximant [4, 4, 3] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.5902 + 0.366 i	0.272 - 0.0715 i
2	1.5902 - 0.366 i	0.272 + 0.0715 i
3	2.346	0.97
4	-3.8027	0.026
5	-4.9206	0.0275 i
6	-4.6869 + 7.0614 i	0.0772 - 0.0877 i
7	-4.6869 - 7.0614 i	0.0772 + 0.0877 i
8	33.7868	5.52 i

Table 12. Singularities with their weights for the quadratic approximant [4, 4, 4] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.5912 + 0.3735 i	0.27 - 0.029 i
2	1.5912 - 0.3735 i	0.27 + 0.029 i
3	2.248	0.772
4	-0.6592 + 2.4583 i	0.00443 + 0.0109 i
5	-0.6592 - 2.4583 i	0.00443 - 0.0109 i
6	-0.679 + 2.5177 i	0.0113 - 0.00419 i
7	-0.679 - 2.5177 i	0.0113 + 0.00419 i
8	-4.9634	0.0503

Table 13. Singularities with their weights for the quadratic approximant [5, 4, 4] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.5787 + 0.2702 i	0.0797 + 0.372 i
2	1.5787 - 0.2702 i	0.0797 - 0.372 i
3	2.2061 + 0.727 i	0.224 + 0.211 i
4	2.2061 - 0.727 i	0.224 - 0.211 i
5	-2.4712 + 0.0271 i	0.00306 + 0.00308 i
6	-2.4712 - 0.0271 i	0.00306 - 0.00308 i
7	3.2013	0.561
8	-4.9032	0.0325
9	16.4278	0.564 i

Table 14. Singularities with their weights for the quadratic approximant [5, 5, 4] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.3998 + 0.0329 i	0.0394 - 0.0367 i
2	1.3998 - 0.0329 i	0.0394 + 0.0367 i
3	1.6915 + 0.4481 i	0.0679 + 0.341 i
4	1.6915 - 0.4481 i	0.0679 - 0.341 i
5	2.1807	1.16
6	-3.7952 + 0.0761 i	0.102 + 0.0968 i
7	-3.7952 - 0.0761 i	0.102 - 0.0968 i
8	-9.4334	1.08
9	-17.5377	1.01 i
10	34.1665	1.23 i

Table 15. Singularities with their weights for the quadratic approximant [5, 5, 5] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.4097 + 0.0365 i	0.0416 - 0.038 i
2	1.4097 - 0.0365 i	0.0416 + 0.038 i
3	1.6983 + 0.4525 i	0.0356 + 0.345 i
4	1.6983 - 0.4525 i	0.0356 - 0.345 i
5	2.1646	1.15
6	-3.8295 + 0.0977 i	0.0787 + 0.0736 i
7	-3.8295 - 0.0977 i	0.0787 - 0.0736 i
8	-10.7667 + 2.519 i	0.434 + 1.18 i
9	-10.7667 - 2.519 i	0.434 - 1.18 i
10	66.1385	0.427 i

Table 16. Singularities with their weights for the quadratic approximant [6, 5, 5] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.4258 + 0.0499 i	0.038 - 0.0309 i
2	1.4258 - 0.0499 i	0.038 + 0.0309 i
3	1.7378 + 0.4815 i	0.17 - 0.267 i
4	1.7378 - 0.4815 i	0.17 + 0.267 i
5	1.9955	0.743
6	-3.3634 + 0.2346 i	0.00327 + 0.00418 i
7	-3.3634 - 0.2346 i	0.00327 - 0.00418 i
8	-4.2643	0.00754
9	-1.0869 + 10.6297 i	0.144 + 0.088 i
10	-1.0869 - 10.6297 i	0.144 - 0.088 i
11	-13.2713	0.121 i

Table 17. Singularities with their weights for the quadratic approximant [6, 6, 5] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.4444 + 0.0692 i	0.039 - 0.0286 i
2	1.4444 - 0.0692 i	0.039 + 0.0286 i
3	1.7233 + 0.5483 i	0.108 - 0.124 i
4	1.7233 - 0.5483 i	0.108 + 0.124 i
5	1.8216	0.528
6	2.862	0.678 i
7	-3.6866 + 0.2224 i	0.015 + 0.0146 i
8	-3.6866 - 0.2224 i	0.015 - 0.0146 i
9	3.7212	1.25
10	-6.8583	0.086
11	18.0958	2.98 i
12	-27.5978	1.28 i

Table 18. Singularities with their weights for the quadratic approximant [6, 6, 6] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	0.3727 + 0.e-5 i	1.44e-7 - 1.44e-7 i
2	0.3727 - 0.e-5 i	1.44e-7 + 1.44e-7 i
3	1.1773 + 0.0065 i	0.000619 - 0.0006 i
4	1.1773 - 0.0065 i	0.000619 + 0.0006 i
5	1.4178	0.0127
6	1.6929 + 0.6798 i	0.0406 - 0.00596 i
7	1.6929 - 0.6798 i	0.0406 + 0.00596 i
8	2.8668	0.496 i
9	-3.44	0.00855
10	-4.4027	0.0123 i
11	4.9751	1.09
12	-14.3294	0.222

Table 19. Singularities with their weights for the quadratic approximant [7, 6, 6] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.3452 + 0.1314 i	0.0405 + 0.0124 i
2	1.3452 - 0.1314 i	0.0405 - 0.0124 i
3	1.3586 + 0.107 i	0.0153 - 0.0388 i
4	1.3586 - 0.107 i	0.0153 + 0.0388 i
5	1.6981 + 0.4596 i	0.0258 - 0.341 i
6	1.6981 - 0.4596 i	0.0258 + 0.341 i
7	2.0846	0.786
8	-3.609 + 0.2265 i	0.0104 + 0.0114 i
9	-3.609 - 0.2265 i	0.0104 - 0.0114 i
10	-5.6194	0.0335
11	-0.0007 + 13.8331 i	0.235 + 0.203 i
12	-0.0007 - 13.8331 i	0.235 - 0.203 i
13	-35.1917	1.33 i

Table 20. Singularities with their weights for the quadratic approximant [7, 7, 6] The most stable singularity is highlighted.
No.	z_c	c	Location in the complex plane
1	1.4624	0.0386
2	1.4272 + 0.3745 i	0.0121 + 0.00503 i
3	1.4272 - 0.3745 i	0.0121 - 0.00503 i
4	1.4101 + 0.439 i	0.00685 - 0.0103 i
5	1.4101 - 0.439 i	0.00685 + 0.0103 i
6	1.9162 + 0.6441 i	0.0472 + 0.0992 i
7	1.9162 - 0.6441 i	0.0472 - 0.0992 i
8	-3.3868	0.00702
9	-4.5445	0.00859 i
10	-5.7371 + 4.805 i	0.00142 - 0.0249 i
11	-5.7371 - 4.805 i	0.00142 + 0.0249 i
12	-4.0523 + 7.3331 i	0.0313 - 0.0184 i
13	-4.0523 - 7.3331 i	0.0313 + 0.0184 i
14	14.9408	13.8 i