Singularities of Møller-Plesset series: example "a2"

[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

0.8512

0.103
Singularities of quadratic [1, 0, 0] approximant

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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

1.5954

0.452

2

11.7124

1.23 i

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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

1.4546

0.277
Singularities of quadratic [1, 1, 1] approximant

2

6.8478

56.2 i

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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

1.4179

0.239
Singularities of quadratic [2, 1, 1] approximant

2

5.4517

39.3 i

3

188.4068

0.414

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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

-0.688 + 0.0022 i

0.00464 + 0.00462 i
Singularities of quadratic [2, 2, 1] approximant

2

-0.688 - 0.0022 i

0.00464 - 0.00462 i

3

1.4056

0.2

4

8.022

3.22 i

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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

1.6081 + 0.4805 i

0.214 - 0.0025 i
Singularities of quadratic [2, 2, 2] approximant

2

1.6081 - 0.4805 i

0.214 + 0.0025 i

3

3.9297

2.68

4

-20.0858

0.0445

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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

1.4551 + 0.2861 i

0.115 + 0.309 i
Singularities of quadratic [3, 2, 2] approximant

2

1.4551 - 0.2861 i

0.115 - 0.309 i

3

1.7774

0.219

4

4.0396

2.58 i

5

84.5313

0.374

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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

1.5257 + 0.4801 i

0.166 - 0.0301 i
Singularities of quadratic [3, 3, 2] approximant

2

1.5257 - 0.4801 i

0.166 + 0.0301 i

3

2.8473

3.37

4

-6.4736

0.0438

5

-9.4952

0.0509 i

6

21.9312

0.237 i

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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

1.5258 + 0.4568 i

0.179 - 0.0681 i
Singularities of quadratic [3, 3, 3] approximant

2

1.5258 - 0.4568 i

0.179 + 0.0681 i

3

2.7826

1.81

4

-5.734 + 0.526 i

0.0279 + 0.0367 i

5

-5.734 - 0.526 i

0.0279 - 0.0367 i

6

-109.5278

0.0208

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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

1.5261 + 0.4416 i

0.178 - 0.102 i
Singularities of quadratic [4, 3, 3] approximant

2

1.5261 - 0.4416 i

0.178 + 0.102 i

3

2.8706

2.12

4

-5.7987 + 2.978 i

0.00918 + 0.017 i

5

-5.7987 - 2.978 i

0.00918 - 0.017 i

6

-4.9795 + 8.4005 i

0.0258 - 0.00254 i

7

-4.9795 - 8.4005 i

0.0258 + 0.00254 i

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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

0.8872

0.00543
Singularities of quadratic [4, 4, 3] approximant

2

0.888

0.00545 i

3

1.5191 + 0.4597 i

0.148 - 0.0678 i

4

1.5191 - 0.4597 i

0.148 + 0.0678 i

5

2.9837

6.18

6

-5.6562

0.0296

7

-8.0078

0.0341 i

8

28.8515

0.208 i

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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

1.6588 + 0.1495 i

0.213 - 0.189 i
Singularities of quadratic [4, 4, 4] approximant

2

1.6588 - 0.1495 i

0.213 + 0.189 i

3

1.6752 + 0.4066 i

5.96 + 0.157 i

4

1.6752 - 0.4066 i

5.96 - 0.157 i

5

2.2505

0.469

6

-5.4088 + 1.1147 i

0.00764 + 0.00951 i

7

-5.4088 - 1.1147 i

0.00764 - 0.00951 i

8

-22.0942

0.0145

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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

1.4557 + 0.3185 i

0.056 + 0.0409 i
Singularities of quadratic [5, 4, 4] approximant

2

1.4557 - 0.3185 i

0.056 - 0.0409 i

3

1.8667

20.3

4

-2.0364

0.000299

5

-2.068

0.000296 i

6

1.9309 + 0.9203 i

0.0445 + 0.0709 i

7

1.9309 - 0.9203 i

0.0445 - 0.0709 i

8

-3.9059

0.00199

9

-10.1909

0.00645 i

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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

-0.6632

0.0000124
Singularities of quadratic [5, 5, 4] approximant

2

-0.6632

0.0000124 i

3

1.4547 + 0.261 i

0.0527 - 0.00204 i

4

1.4547 - 0.261 i

0.0527 + 0.00204 i

5

1.5661

0.0853

6

1.9399 + 0.7275 i

0.0934 + 0.0925 i

7

1.9399 - 0.7275 i

0.0934 - 0.0925 i

8

-4.6419

0.00602

9

-9.8258

0.0108 i

10

153.4361

0.121 i

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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

-1.2016 + 0.7873 i

8.7e-6 - 0.0000606 i
Singularities of quadratic [5, 5, 5] approximant

2

-1.2016 - 0.7873 i

8.7e-6 + 0.0000606 i

3

-1.2071 + 0.7883 i

0.0000605 + 8.98e-6 i

4

-1.2071 - 0.7883 i

0.0000605 - 8.98e-6 i

5

1.6615 + 0.4532 i

0.128 - 0.403 i

6

1.6615 - 0.4532 i

0.128 + 0.403 i

7

2.3269 + 0.2427 i

0.144 - 0.0854 i

8

2.3269 - 0.2427 i

0.144 + 0.0854 i

9

-4.0348

0.00149

10

4.655

2.9

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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

-0.9175 + 0.6593 i

9.69e-7 + 0.0000281 i
Singularities of quadratic [6, 5, 5] approximant

2

-0.9175 - 0.6593 i

9.69e-7 - 0.0000281 i

3

-0.9185 + 0.6597 i

0.0000281 - 9.38e-7 i

4

-0.9185 - 0.6597 i

0.0000281 + 9.38e-7 i

5

1.6543 + 0.429 i

0.165 - 0.714 i

6

1.6543 - 0.429 i

0.165 + 0.714 i

7

2.3004 + 0.4493 i

0.202 - 0.0467 i

8

2.3004 - 0.4493 i

0.202 + 0.0467 i

9

3.9018

1.7

10

-4.0819

0.00187

11

-28.9909

0.0107 i

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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

-0.8853 + 0.457 i

4.95e-6 - 9.02e-6 i
Singularities of quadratic [6, 6, 5] approximant

2

-0.8853 - 0.457 i

4.95e-6 + 9.02e-6 i

3

-0.8858 + 0.4565 i

9.01e-6 + 4.95e-6 i

4

-0.8858 - 0.4565 i

9.01e-6 - 4.95e-6 i

5

1.6751 + 0.4376 i

0.44 - 0.408 i

6

1.6751 - 0.4376 i

0.44 + 0.408 i

7

2.6201 + 0.2022 i

0.181 - 0.0689 i

8

2.6201 - 0.2022 i

0.181 + 0.0689 i

9

-3.7609

0.00116

10

9.1432

0.0847

11

18.8525

0.0226 i

12

-27.826

0.00958 i

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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

0.3857

2.48e-7
Singularities of quadratic [6, 6, 6] approximant

2

0.3857

2.48e-7 i

3

-0.8874 + 0.5691 i

5.01e-7 + 8.77e-6 i

4

-0.8874 - 0.5691 i

5.01e-7 - 8.77e-6 i

5

-0.8886 + 0.5689 i

8.77e-6 - 4.86e-7 i

6

-0.8886 - 0.5689 i

8.77e-6 + 4.86e-7 i

7

1.5704 + 0.4368 i

0.136 - 0.0324 i

8

1.5704 - 0.4368 i

0.136 + 0.0324 i

9

1.8612 + 0.3125 i

0.148 - 1.11 i

10

1.8612 - 0.3125 i

0.148 + 1.11 i

11

3.3356

2.38

12

-3.8191

0.00109

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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

0.6028

0.0000198
Singularities of quadratic [7, 6, 6] approximant

2

0.6028

0.0000198 i

3

-0.8783 + 0.6021 i

5.72e-7 + 0.0000177 i

4

-0.8783 - 0.6021 i

5.72e-7 - 0.0000177 i

5

-0.8791 + 0.6023 i

0.0000177 - 5.55e-7 i

6

-0.8791 - 0.6023 i

0.0000177 + 5.55e-7 i

7

1.6343 + 0.4285 i

0.225 + 0.555 i

8

1.6343 - 0.4285 i

0.225 - 0.555 i

9

2.1921 + 0.4677 i

0.203 - 0.0835 i

10

2.1921 - 0.4677 i

0.203 + 0.0835 i

11

3.558

1.5

12

-4.0246

0.00169

13

-31.172

0.011 i

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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

0.2425 + 0.e-4 i

9.4e-10 - 9.4e-10 i
Singularities of quadratic [7, 7, 6] approximant

2

0.2425 - 0.e-4 i

9.4e-10 + 9.4e-10 i

3

0.4473 + 0.e-5 i

6.42e-8 - 6.42e-8 i

4

0.4473 - 0.e-5 i

6.42e-8 + 6.42e-8 i

5

-0.9023 + 0.5607 i

8.51e-7 + 3.35e-6 i

6

-0.9023 - 0.5607 i

8.51e-7 - 3.35e-6 i

7

-0.9045 + 0.5596 i

3.35e-6 - 8.36e-7 i

8

-0.9045 - 0.5596 i

3.35e-6 + 8.36e-7 i

9

1.6716 + 0.2972 i

0.437 + 0.205 i

10

1.6716 - 0.2972 i

0.437 - 0.205 i

11

2.8576

5.16

12

-3.5732

0.000582

13

20.867

0.0651 i

14

-217.8922

0.0662 i

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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	0.8512	0.103

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	1.5954	0.452
2	11.7124	1.23 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	1.4546	0.277
2	6.8478	56.2 i

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	1.4179	0.239
2	5.4517	39.3 i
3	188.4068	0.414

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	-0.688 + 0.0022 i	0.00464 + 0.00462 i
2	-0.688 - 0.0022 i	0.00464 - 0.00462 i
3	1.4056	0.2
4	8.022	3.22 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	1.6081 + 0.4805 i	0.214 - 0.0025 i
2	1.6081 - 0.4805 i	0.214 + 0.0025 i
3	3.9297	2.68
4	-20.0858	0.0445

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	1.4551 + 0.2861 i	0.115 + 0.309 i
2	1.4551 - 0.2861 i	0.115 - 0.309 i
3	1.7774	0.219
4	4.0396	2.58 i
5	84.5313	0.374

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	1.5257 + 0.4801 i	0.166 - 0.0301 i
2	1.5257 - 0.4801 i	0.166 + 0.0301 i
3	2.8473	3.37
4	-6.4736	0.0438
5	-9.4952	0.0509 i
6	21.9312	0.237 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	1.5258 + 0.4568 i	0.179 - 0.0681 i
2	1.5258 - 0.4568 i	0.179 + 0.0681 i
3	2.7826	1.81
4	-5.734 + 0.526 i	0.0279 + 0.0367 i
5	-5.734 - 0.526 i	0.0279 - 0.0367 i
6	-109.5278	0.0208

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	1.5261 + 0.4416 i	0.178 - 0.102 i
2	1.5261 - 0.4416 i	0.178 + 0.102 i
3	2.8706	2.12
4	-5.7987 + 2.978 i	0.00918 + 0.017 i
5	-5.7987 - 2.978 i	0.00918 - 0.017 i
6	-4.9795 + 8.4005 i	0.0258 - 0.00254 i
7	-4.9795 - 8.4005 i	0.0258 + 0.00254 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	0.8872	0.00543
2	0.888	0.00545 i
3	1.5191 + 0.4597 i	0.148 - 0.0678 i
4	1.5191 - 0.4597 i	0.148 + 0.0678 i
5	2.9837	6.18
6	-5.6562	0.0296
7	-8.0078	0.0341 i
8	28.8515	0.208 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	1.6588 + 0.1495 i	0.213 - 0.189 i
2	1.6588 - 0.1495 i	0.213 + 0.189 i
3	1.6752 + 0.4066 i	5.96 + 0.157 i
4	1.6752 - 0.4066 i	5.96 - 0.157 i
5	2.2505	0.469
6	-5.4088 + 1.1147 i	0.00764 + 0.00951 i
7	-5.4088 - 1.1147 i	0.00764 - 0.00951 i
8	-22.0942	0.0145

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	1.4557 + 0.3185 i	0.056 + 0.0409 i
2	1.4557 - 0.3185 i	0.056 - 0.0409 i
3	1.8667	20.3
4	-2.0364	0.000299
5	-2.068	0.000296 i
6	1.9309 + 0.9203 i	0.0445 + 0.0709 i
7	1.9309 - 0.9203 i	0.0445 - 0.0709 i
8	-3.9059	0.00199
9	-10.1909	0.00645 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	-0.6632	0.0000124
2	-0.6632	0.0000124 i
3	1.4547 + 0.261 i	0.0527 - 0.00204 i
4	1.4547 - 0.261 i	0.0527 + 0.00204 i
5	1.5661	0.0853
6	1.9399 + 0.7275 i	0.0934 + 0.0925 i
7	1.9399 - 0.7275 i	0.0934 - 0.0925 i
8	-4.6419	0.00602
9	-9.8258	0.0108 i
10	153.4361	0.121 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	-1.2016 + 0.7873 i	8.7e-6 - 0.0000606 i
2	-1.2016 - 0.7873 i	8.7e-6 + 0.0000606 i
3	-1.2071 + 0.7883 i	0.0000605 + 8.98e-6 i
4	-1.2071 - 0.7883 i	0.0000605 - 8.98e-6 i
5	1.6615 + 0.4532 i	0.128 - 0.403 i
6	1.6615 - 0.4532 i	0.128 + 0.403 i
7	2.3269 + 0.2427 i	0.144 - 0.0854 i
8	2.3269 - 0.2427 i	0.144 + 0.0854 i
9	-4.0348	0.00149
10	4.655	2.9

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	-0.9175 + 0.6593 i	9.69e-7 + 0.0000281 i
2	-0.9175 - 0.6593 i	9.69e-7 - 0.0000281 i
3	-0.9185 + 0.6597 i	0.0000281 - 9.38e-7 i
4	-0.9185 - 0.6597 i	0.0000281 + 9.38e-7 i
5	1.6543 + 0.429 i	0.165 - 0.714 i
6	1.6543 - 0.429 i	0.165 + 0.714 i
7	2.3004 + 0.4493 i	0.202 - 0.0467 i
8	2.3004 - 0.4493 i	0.202 + 0.0467 i
9	3.9018	1.7
10	-4.0819	0.00187
11	-28.9909	0.0107 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	-0.8853 + 0.457 i	4.95e-6 - 9.02e-6 i
2	-0.8853 - 0.457 i	4.95e-6 + 9.02e-6 i
3	-0.8858 + 0.4565 i	9.01e-6 + 4.95e-6 i
4	-0.8858 - 0.4565 i	9.01e-6 - 4.95e-6 i
5	1.6751 + 0.4376 i	0.44 - 0.408 i
6	1.6751 - 0.4376 i	0.44 + 0.408 i
7	2.6201 + 0.2022 i	0.181 - 0.0689 i
8	2.6201 - 0.2022 i	0.181 + 0.0689 i
9	-3.7609	0.00116
10	9.1432	0.0847
11	18.8525	0.0226 i
12	-27.826	0.00958 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	0.3857	2.48e-7
2	0.3857	2.48e-7 i
3	-0.8874 + 0.5691 i	5.01e-7 + 8.77e-6 i
4	-0.8874 - 0.5691 i	5.01e-7 - 8.77e-6 i
5	-0.8886 + 0.5689 i	8.77e-6 - 4.86e-7 i
6	-0.8886 - 0.5689 i	8.77e-6 + 4.86e-7 i
7	1.5704 + 0.4368 i	0.136 - 0.0324 i
8	1.5704 - 0.4368 i	0.136 + 0.0324 i
9	1.8612 + 0.3125 i	0.148 - 1.11 i
10	1.8612 - 0.3125 i	0.148 + 1.11 i
11	3.3356	2.38
12	-3.8191	0.00109

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	0.6028	0.0000198
2	0.6028	0.0000198 i
3	-0.8783 + 0.6021 i	5.72e-7 + 0.0000177 i
4	-0.8783 - 0.6021 i	5.72e-7 - 0.0000177 i
5	-0.8791 + 0.6023 i	0.0000177 - 5.55e-7 i
6	-0.8791 - 0.6023 i	0.0000177 + 5.55e-7 i
7	1.6343 + 0.4285 i	0.225 + 0.555 i
8	1.6343 - 0.4285 i	0.225 - 0.555 i
9	2.1921 + 0.4677 i	0.203 - 0.0835 i
10	2.1921 - 0.4677 i	0.203 + 0.0835 i
11	3.558	1.5
12	-4.0246	0.00169
13	-31.172	0.011 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	0.2425 + 0.e-4 i	9.4e-10 - 9.4e-10 i
2	0.2425 - 0.e-4 i	9.4e-10 + 9.4e-10 i
3	0.4473 + 0.e-5 i	6.42e-8 - 6.42e-8 i
4	0.4473 - 0.e-5 i	6.42e-8 + 6.42e-8 i
5	-0.9023 + 0.5607 i	8.51e-7 + 3.35e-6 i
6	-0.9023 - 0.5607 i	8.51e-7 - 3.35e-6 i
7	-0.9045 + 0.5596 i	3.35e-6 - 8.36e-7 i
8	-0.9045 - 0.5596 i	3.35e-6 + 8.36e-7 i
9	1.6716 + 0.2972 i	0.437 + 0.205 i
10	1.6716 - 0.2972 i	0.437 - 0.205 i
11	2.8576	5.16
12	-3.5732	0.000582
13	20.867	0.0651 i
14	-217.8922	0.0662 i