Singularities of Møller-Plesset series: example "a38"

[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

1.3265

0.209
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

1.2654

0.19

2

2330.1933

8.15 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

1.5034

0.366
Singularities of quadratic [1, 1, 1] approximant

2

-8.019

0.922

  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

1.4651

0.327
Singularities of quadratic [2, 1, 1] approximant

2

-4.851

0.591

3

-10.0544

0.378 i

  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

1.4514

0.304
Singularities of quadratic [2, 2, 1] approximant

2

-5.9008

1.21

3

-12.6702

0.548 i

4

409.7523

2.76 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

1.4865

0.397
Singularities of quadratic [2, 2, 2] approximant

2

4.1292

0.718 i

3

-5.6066

0.454

4

5.8887

2.52

  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

0.8396

0.0291
Singularities of quadratic [3, 2, 2] approximant

2

0.8483

0.0289 i

3

1.3991

0.192

4

-5.1956

0.442

5

-18.713

0.365 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

1.5944 + 0.3268 i

0.312 - 0.153 i
Singularities of quadratic [3, 3, 2] approximant

2

1.5944 - 0.3268 i

0.312 + 0.153 i

3

2.3197

0.851

4

-3.8451

0.119

5

-12.803

0.179 i

6

18.0688

0.357 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

1.5886 + 0.3256 i

0.304 - 0.16 i
Singularities of quadratic [3, 3, 3] approximant

2

1.5886 - 0.3256 i

0.304 + 0.16 i

3

2.2805

0.759

4

-3.8327

0.119

5

-11.7935

0.176 i

6

21.4977

0.352 i

  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

1.6162 + 0.3654 i

0.274 - 0.0141 i
Singularities of quadratic [4, 3, 3] approximant

2

1.6162 - 0.3654 i

0.274 + 0.0141 i

3

2.6238

5.7

4

-3.5012

0.062

5

-1.5659 + 8.783 i

0.0332 + 0.119 i

6

-1.5659 - 8.783 i

0.0332 - 0.119 i

7

-76.3262

0.878 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

1.6056 + 0.3443 i

0.306 - 0.0756 i
Singularities of quadratic [4, 4, 3] approximant

2

1.6056 - 0.3443 i

0.306 + 0.0756 i

3

-1.8411

0.011

4

-1.8583

0.0111 i

5

2.4182

1.38

6

-4.2886

0.357

7

-10.0149

0.178 i

8

22.3904

0.368 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

1.5973 + 0.3038 i

0.302 - 0.285 i
Singularities of quadratic [4, 4, 4] approximant

2

1.5973 - 0.3038 i

0.302 + 0.285 i

3

-2.0892 + 0.0178 i

0.00762 + 0.00774 i

4

-2.0892 - 0.0178 i

0.00762 - 0.00774 i

5

2.4079

0.799

6

-3.694

0.0586

7

5.9493

0.767 i

8

31.8549

1.46

  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

1.5942 + 0.2933 i

0.255 - 0.347 i
Singularities of quadratic [5, 4, 4] approximant

2

1.5942 - 0.2933 i

0.255 + 0.347 i

3

-1.762 + 0.0075 i

0.00312 + 0.00313 i

4

-1.762 - 0.0075 i

0.00312 - 0.00313 i

5

2.4553

0.8

6

-3.5267

0.045

7

5.3396

0.851 i

8

11.14 + 15.0748 i

0.408 + 0.405 i

9

11.14 - 15.0748 i

0.408 - 0.405 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

1.3587 + 0.0095 i

0.081 - 0.0901 i
Singularities of quadratic [5, 5, 4] approximant

2

1.3587 - 0.0095 i

0.081 + 0.0901 i

3

1.6448 + 0.3601 i

0.345 + 0.0715 i

4

1.6448 - 0.3601 i

0.345 - 0.0715 i

5

-2.3256

0.0183

6

-2.4077

0.0187 i

7

2.5195

2.92

8

-4.9147

5.48

9

-8.3908

0.192 i

10

26.575

0.393 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

1.0025 + 0.0001 i

0.0295 - 0.0297 i
Singularities of quadratic [5, 5, 5] approximant

2

1.0025 - 0.0001 i

0.0295 + 0.0297 i

3

1.6074 + 0.3297 i

0.334 - 0.135 i

4

1.6074 - 0.3297 i

0.334 + 0.135 i

5

-2.1751 + 0.0114 i

0.0271 + 0.0282 i

6

-2.1751 - 0.0114 i

0.0271 - 0.0282 i

7

2.4342

1.17

8

-3.9298

0.103

9

9.0674

0.425 i

10

-44.7208

0.208 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

1.4724 + 0.227 i

0.0412 + 0.0306 i
Singularities of quadratic [6, 5, 5] approximant

2

1.4724 - 0.227 i

0.0412 - 0.0306 i

3

1.399 + 0.7826 i

0.011 - 0.00462 i

4

1.399 - 0.7826 i

0.011 + 0.00462 i

5

1.4898 + 0.7394 i

0.00664 + 0.0123 i

6

1.4898 - 0.7394 i

0.00664 - 0.0123 i

7

-2.0214 + 0.0344 i

0.00182 + 0.0018 i

8

-2.0214 - 0.0344 i

0.00182 - 0.0018 i

9

-3.3883

0.0203

10

4.4825

0.416

11

-12.1587

0.763 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

1.4792 + 0.246 i

0.0351 + 0.0418 i
Singularities of quadratic [6, 6, 5] approximant

2

1.4792 - 0.246 i

0.0351 - 0.0418 i

3

1.3152 + 0.7784 i

0.00715 - 0.00355 i

4

1.3152 - 0.7784 i

0.00715 + 0.00355 i

5

1.3682 + 0.7509 i

0.00435 + 0.00746 i

6

1.3682 - 0.7509 i

0.00435 - 0.00746 i

7

-2.0343 + 0.0358 i

0.00189 + 0.00187 i

8

-2.0343 - 0.0358 i

0.00189 - 0.00187 i

9

-3.4166

0.0214

10

4.0852

0.553

11

-12.0568

0.653 i

12

2842.6613

8.3 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

0.3117

1.25e-7
Singularities of quadratic [6, 6, 6] approximant

2

0.3117

1.25e-7 i

3

1.3571 + 0.2802 i

0.0043 + 0.00579 i

4

1.3571 - 0.2802 i

0.0043 - 0.00579 i

5

1.347 + 0.6345 i

0.00201 + 0.00264 i

6

1.347 - 0.6345 i

0.00201 - 0.00264 i

7

1.2963 + 0.7464 i

0.00312 - 0.000908 i

8

1.2963 - 0.7464 i

0.00312 + 0.000908 i

9

-2.046

0.00278

10

-2.1349

0.00306 i

11

-3.9782

0.282

12

6.8139

0.421

  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

1.3403

0.00869
Singularities of quadratic [7, 6, 6] approximant

2

0.7509 + 1.26 i

0.000301 + 0.000151 i

3

0.7509 - 1.26 i

0.000301 - 0.000151 i

4

0.7507 + 1.282 i

0.000158 - 0.000303 i

5

0.7507 - 1.282 i

0.000158 + 0.000303 i

6

1.4416 + 0.5718 i

0.00304 + 0.00613 i

7

1.4416 - 0.5718 i

0.00304 - 0.00613 i

8

1.899

0.457 i

9

-2.0796

0.00125

10

-2.313

0.00154 i

11

1.6087 + 3.981 i

0.00283 - 0.0166 i

12

1.6087 - 3.981 i

0.00283 + 0.0166 i

13

-7.842

0.0545

  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

-0.4363 + 0.e-4 i

3.44e-8 + 3.44e-8 i
Singularities of quadratic [7, 7, 6] approximant

2

-0.4363 - 0.e-4 i

3.44e-8 - 3.44e-8 i

3

0.9417 + 0.0008 i

0.000044 - 0.0000438 i

4

0.9417 - 0.0008 i

0.000044 + 0.0000438 i

5

1.4007

0.0119

6

-0.5477 + 1.7257 i

0.000157 - 0.0000204 i

7

-0.5477 - 1.7257 i

0.000157 + 0.0000204 i

8

-0.5898 + 1.7378 i

0.0000108 + 0.00016 i

9

-0.5898 - 1.7378 i

0.0000108 - 0.00016 i

10

1.753 + 0.7754 i

0.0184 + 0.0153 i

11

1.753 - 0.7754 i

0.0184 - 0.0153 i

12

-2.0694

0.000459

13

-2.8194

0.00141 i

14

86.723

0.76 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	1.3265	0.209

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	1.2654	0.19
2	2330.1933	8.15 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	1.5034	0.366
2	-8.019	0.922

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	1.4651	0.327
2	-4.851	0.591
3	-10.0544	0.378 i

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	1.4514	0.304
2	-5.9008	1.21
3	-12.6702	0.548 i
4	409.7523	2.76 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	1.4865	0.397
2	4.1292	0.718 i
3	-5.6066	0.454
4	5.8887	2.52

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	0.8396	0.0291
2	0.8483	0.0289 i
3	1.3991	0.192
4	-5.1956	0.442
5	-18.713	0.365 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	1.5944 + 0.3268 i	0.312 - 0.153 i
2	1.5944 - 0.3268 i	0.312 + 0.153 i
3	2.3197	0.851
4	-3.8451	0.119
5	-12.803	0.179 i
6	18.0688	0.357 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	1.5886 + 0.3256 i	0.304 - 0.16 i
2	1.5886 - 0.3256 i	0.304 + 0.16 i
3	2.2805	0.759
4	-3.8327	0.119
5	-11.7935	0.176 i
6	21.4977	0.352 i

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	1.6162 + 0.3654 i	0.274 - 0.0141 i
2	1.6162 - 0.3654 i	0.274 + 0.0141 i
3	2.6238	5.7
4	-3.5012	0.062
5	-1.5659 + 8.783 i	0.0332 + 0.119 i
6	-1.5659 - 8.783 i	0.0332 - 0.119 i
7	-76.3262	0.878 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	1.6056 + 0.3443 i	0.306 - 0.0756 i
2	1.6056 - 0.3443 i	0.306 + 0.0756 i
3	-1.8411	0.011
4	-1.8583	0.0111 i
5	2.4182	1.38
6	-4.2886	0.357
7	-10.0149	0.178 i
8	22.3904	0.368 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	1.5973 + 0.3038 i	0.302 - 0.285 i
2	1.5973 - 0.3038 i	0.302 + 0.285 i
3	-2.0892 + 0.0178 i	0.00762 + 0.00774 i
4	-2.0892 - 0.0178 i	0.00762 - 0.00774 i
5	2.4079	0.799
6	-3.694	0.0586
7	5.9493	0.767 i
8	31.8549	1.46

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	1.5942 + 0.2933 i	0.255 - 0.347 i
2	1.5942 - 0.2933 i	0.255 + 0.347 i
3	-1.762 + 0.0075 i	0.00312 + 0.00313 i
4	-1.762 - 0.0075 i	0.00312 - 0.00313 i
5	2.4553	0.8
6	-3.5267	0.045
7	5.3396	0.851 i
8	11.14 + 15.0748 i	0.408 + 0.405 i
9	11.14 - 15.0748 i	0.408 - 0.405 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	1.3587 + 0.0095 i	0.081 - 0.0901 i
2	1.3587 - 0.0095 i	0.081 + 0.0901 i
3	1.6448 + 0.3601 i	0.345 + 0.0715 i
4	1.6448 - 0.3601 i	0.345 - 0.0715 i
5	-2.3256	0.0183
6	-2.4077	0.0187 i
7	2.5195	2.92
8	-4.9147	5.48
9	-8.3908	0.192 i
10	26.575	0.393 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	1.0025 + 0.0001 i	0.0295 - 0.0297 i
2	1.0025 - 0.0001 i	0.0295 + 0.0297 i
3	1.6074 + 0.3297 i	0.334 - 0.135 i
4	1.6074 - 0.3297 i	0.334 + 0.135 i
5	-2.1751 + 0.0114 i	0.0271 + 0.0282 i
6	-2.1751 - 0.0114 i	0.0271 - 0.0282 i
7	2.4342	1.17
8	-3.9298	0.103
9	9.0674	0.425 i
10	-44.7208	0.208 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	1.4724 + 0.227 i	0.0412 + 0.0306 i
2	1.4724 - 0.227 i	0.0412 - 0.0306 i
3	1.399 + 0.7826 i	0.011 - 0.00462 i
4	1.399 - 0.7826 i	0.011 + 0.00462 i
5	1.4898 + 0.7394 i	0.00664 + 0.0123 i
6	1.4898 - 0.7394 i	0.00664 - 0.0123 i
7	-2.0214 + 0.0344 i	0.00182 + 0.0018 i
8	-2.0214 - 0.0344 i	0.00182 - 0.0018 i
9	-3.3883	0.0203
10	4.4825	0.416
11	-12.1587	0.763 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	1.4792 + 0.246 i	0.0351 + 0.0418 i
2	1.4792 - 0.246 i	0.0351 - 0.0418 i
3	1.3152 + 0.7784 i	0.00715 - 0.00355 i
4	1.3152 - 0.7784 i	0.00715 + 0.00355 i
5	1.3682 + 0.7509 i	0.00435 + 0.00746 i
6	1.3682 - 0.7509 i	0.00435 - 0.00746 i
7	-2.0343 + 0.0358 i	0.00189 + 0.00187 i
8	-2.0343 - 0.0358 i	0.00189 - 0.00187 i
9	-3.4166	0.0214
10	4.0852	0.553
11	-12.0568	0.653 i
12	2842.6613	8.3 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	0.3117	1.25e-7
2	0.3117	1.25e-7 i
3	1.3571 + 0.2802 i	0.0043 + 0.00579 i
4	1.3571 - 0.2802 i	0.0043 - 0.00579 i
5	1.347 + 0.6345 i	0.00201 + 0.00264 i
6	1.347 - 0.6345 i	0.00201 - 0.00264 i
7	1.2963 + 0.7464 i	0.00312 - 0.000908 i
8	1.2963 - 0.7464 i	0.00312 + 0.000908 i
9	-2.046	0.00278
10	-2.1349	0.00306 i
11	-3.9782	0.282
12	6.8139	0.421

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	1.3403	0.00869
2	0.7509 + 1.26 i	0.000301 + 0.000151 i
3	0.7509 - 1.26 i	0.000301 - 0.000151 i
4	0.7507 + 1.282 i	0.000158 - 0.000303 i
5	0.7507 - 1.282 i	0.000158 + 0.000303 i
6	1.4416 + 0.5718 i	0.00304 + 0.00613 i
7	1.4416 - 0.5718 i	0.00304 - 0.00613 i
8	1.899	0.457 i
9	-2.0796	0.00125
10	-2.313	0.00154 i
11	1.6087 + 3.981 i	0.00283 - 0.0166 i
12	1.6087 - 3.981 i	0.00283 + 0.0166 i
13	-7.842	0.0545

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	-0.4363 + 0.e-4 i	3.44e-8 + 3.44e-8 i
2	-0.4363 - 0.e-4 i	3.44e-8 - 3.44e-8 i
3	0.9417 + 0.0008 i	0.000044 - 0.0000438 i
4	0.9417 - 0.0008 i	0.000044 + 0.0000438 i
5	1.4007	0.0119
6	-0.5477 + 1.7257 i	0.000157 - 0.0000204 i
7	-0.5477 - 1.7257 i	0.000157 + 0.0000204 i
8	-0.5898 + 1.7378 i	0.0000108 + 0.00016 i
9	-0.5898 - 1.7378 i	0.0000108 - 0.00016 i
10	1.753 + 0.7754 i	0.0184 + 0.0153 i
11	1.753 - 0.7754 i	0.0184 - 0.0153 i
12	-2.0694	0.000459
13	-2.8194	0.00141 i
14	86.723	0.76 i