Singularities of Møller-Plesset series: example "a45"

[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

-2.2575

1.41
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

-0.3276

0.0614

2

-0.855

0.0993 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

-0.7732

0.482
Singularities of quadratic [1, 1, 1] approximant

2

1.6974

3.14

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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

-0.9385

1.72
Singularities of quadratic [2, 1, 1] approximant

2

1.4547

0.844

3

-2.7429

0.833 i

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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

1.2488

0.457
Singularities of quadratic [2, 2, 1] approximant

2

-1.2595 + 0.2706 i

1.65 - 1.03 i

3

-1.2595 - 0.2706 i

1.65 + 1.03 i

4

131.6219

4.86 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

-1.1965 + 0.3195 i

1.62 - 0.359 i
Singularities of quadratic [2, 2, 2] approximant

2

-1.1965 - 0.3195 i

1.62 + 0.359 i

3

1.2757

0.499

4

-19.4413

3.56e3

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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

-0.6639

0.082
Singularities of quadratic [3, 2, 2] approximant

2

-0.7987 + 0.0618 i

0.042 - 0.0952 i

3

-0.7987 - 0.0618 i

0.042 + 0.0952 i

4

1.2029

0.341

5

-1.3708

699. i

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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

-0.8297 + 0.4026 i

0.183 + 0.0479 i
Singularities of quadratic [3, 3, 2] approximant

2

-0.8297 - 0.4026 i

0.183 - 0.0479 i

3

1.1669

0.277

4

-1.2035 + 0.4556 i

0.0417 - 0.376 i

5

-1.2035 - 0.4556 i

0.0417 + 0.376 i

6

115.5956

11.7 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

-0.5061 + 0.1173 i

0.00556 + 0.00312 i
Singularities of quadratic [3, 3, 3] approximant

2

-0.5061 - 0.1173 i

0.00556 - 0.00312 i

3

-0.5063 + 0.1547 i

0.00305 - 0.00667 i

4

-0.5063 - 0.1547 i

0.00305 + 0.00667 i

5

1.1235

0.182

6

-3.9631

0.511

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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

-0.9014 + 0.1456 i

0.356 + 0.0481 i
Singularities of quadratic [4, 3, 3] approximant

2

-0.9014 - 0.1456 i

0.356 - 0.0481 i

3

1.2576

0.643

4

-0.1345 + 1.3902 i

0.0492 + 0.122 i

5

-0.1345 - 1.3902 i

0.0492 - 0.122 i

6

-0.4801 + 1.5301 i

0.139 - 0.0342 i

7

-0.4801 - 1.5301 i

0.139 + 0.0342 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

-0.9626 + 0.3154 i

0.526 + 0.791 i
Singularities of quadratic [4, 4, 3] approximant

2

-0.9626 - 0.3154 i

0.526 - 0.791 i

3

1.1948 + 0.3122 i

0.16 - 0.235 i

4

1.1948 - 0.3122 i

0.16 + 0.235 i

5

1.7604

0.316

6

-1.8951

1.43

7

-3.0676

9.28 i

8

14.2401

10.9 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

-0.9127 + 0.2611 i

0.29 - 0.233 i
Singularities of quadratic [4, 4, 4] approximant

2

-0.9127 - 0.2611 i

0.29 + 0.233 i

3

1.2313 + 0.3763 i

0.223 - 0.0876 i

4

1.2313 - 0.3763 i

0.223 + 0.0876 i

5

-1.1708 + 0.6464 i

0.255 - 0.0129 i

6

-1.1708 - 0.6464 i

0.255 + 0.0129 i

7

-1.4972

1.21

8

2.7567

2.68

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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

-0.8854 + 0.2175 i

0.203 + 0.00228 i
Singularities of quadratic [5, 4, 4] approximant

2

-0.8854 - 0.2175 i

0.203 - 0.00228 i

3

-1.0938

1.38

4

-1.0844 + 0.5813 i

0.213 + 0.106 i

5

-1.0844 - 0.5813 i

0.213 - 0.106 i

6

1.2304 + 0.3813 i

0.219 - 0.0733 i

7

1.2304 - 0.3813 i

0.219 + 0.0733 i

8

2.7544

2.99

9

-34.7857

0.542 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

-0.9549 + 0.3315 i

0.597 + 0.532 i
Singularities of quadratic [5, 5, 4] approximant

2

-0.9549 - 0.3315 i

0.597 - 0.532 i

3

1.2329 + 0.3181 i

0.198 - 0.292 i

4

1.2329 - 0.3181 i

0.198 + 0.292 i

5

-1.5857

1.4

6

2.1365

0.447

7

-2.4589

2.93 i

8

5.16

22.4 i

9

-5.4255 + 3.783 i

1.3 + 0.897 i

10

-5.4255 - 3.783 i

1.3 - 0.897 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

-0.955 + 0.3342 i

0.612 + 0.48 i
Singularities of quadratic [5, 5, 5] approximant

2

-0.955 - 0.3342 i

0.612 - 0.48 i

3

1.2313 + 0.323 i

0.201 - 0.269 i

4

1.2313 - 0.323 i

0.201 + 0.269 i

5

-1.5067

1.25

6

2.1916

0.506

7

-3.0299

0.747 i

8

-2.8051 + 3.5101 i

0.411 + 0.406 i

9

-2.8051 - 3.5101 i

0.411 - 0.406 i

10

9.4584

3.76 i

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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

-0.9539 + 0.3289 i

0.545 + 0.568 i
Singularities of quadratic [6, 5, 5] approximant

2

-0.9539 - 0.3289 i

0.545 - 0.568 i

3

1.2216 + 0.3448 i

0.206 - 0.17 i

4

1.2216 - 0.3448 i

0.206 + 0.17 i

5

-1.6333

1.44

6

2.2225

0.776

7

0.8593 + 2.5653 i

0.165 - 0.276 i

8

0.8593 - 2.5653 i

0.165 + 0.276 i

9

0.3064 + 2.9198 i

0.243 + 0.204 i

10

0.3064 - 2.9198 i

0.243 - 0.204 i

11

-3.0106

1.45 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

-0.954 + 0.3294 i

0.554 + 0.56 i
Singularities of quadratic [6, 6, 5] approximant

2

-0.954 - 0.3294 i

0.554 - 0.56 i

3

1.2172 + 0.3488 i

0.195 - 0.154 i

4

1.2172 - 0.3488 i

0.195 + 0.154 i

5

-1.6222

1.42

6

2.3307

1.17

7

1.3788 + 2.1767 i

0.012 - 0.326 i

8

1.3788 - 2.1767 i

0.012 + 0.326 i

9

-2.8981

1.58 i

10

1.581 + 3.4057 i

0.361 + 0.107 i

11

1.581 - 3.4057 i

0.361 - 0.107 i

12

62.3914

1.91 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

-0.9591 + 0.3285 i

0.695 + 0.626 i
Singularities of quadratic [6, 6, 6] approximant

2

-0.9591 - 0.3285 i

0.695 - 0.626 i

3

1.2166 + 0.3354 i

0.127 - 0.214 i

4

1.2166 - 0.3354 i

0.127 + 0.214 i

5

1.1167 + 1.0374 i

0.0617 - 0.0472 i

6

1.1167 - 1.0374 i

0.0617 + 0.0472 i

7

1.1665 + 0.9898 i

0.0485 + 0.059 i

8

1.1665 - 0.9898 i

0.0485 - 0.059 i

9

-1.5281 + 0.3455 i

0.86 - 0.493 i

10

-1.5281 - 0.3455 i

0.86 + 0.493 i

11

-2.1428

0.666

12

3.7492

152.

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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

-0.9562 + 0.3284 i

0.604 + 0.604 i
Singularities of quadratic [7, 6, 6] approximant

2

-0.9562 - 0.3284 i

0.604 - 0.604 i

3

1.2684

0.276

4

1.2969 + 0.2219 i

0.0502 - 0.441 i

5

1.2969 - 0.2219 i

0.0502 + 0.441 i

6

1.2753 + 0.3482 i

0.878 + 1.01 i

7

1.2753 - 0.3482 i

0.878 - 1.01 i

8

-1.8068 + 0.2518 i

1.56 - 1.26 i

9

-1.8068 - 0.2518 i

1.56 + 1.26 i

10

2.3937

0.715 i

11

-5.0215

0.837

12

2.9629 + 4.1011 i

0.411 + 1.15 i

13

2.9629 - 4.1011 i

0.411 - 1.15 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

-0.956 + 0.3284 i

0.598 + 0.602 i
Singularities of quadratic [7, 7, 6] approximant

2

-0.956 - 0.3284 i

0.598 - 0.602 i

3

1.2051 + 0.327 i

0.0953 - 0.274 i

4

1.2051 - 0.327 i

0.0953 + 0.274 i

5

1.4982 + 0.3994 i

0.246 + 0.0779 i

6

1.4982 - 0.3994 i

0.246 - 0.0779 i

7

1.7851

0.5

8

-1.7995 + 0.2582 i

1.54 - 1.17 i

9

-1.7995 - 0.2582 i

1.54 + 1.17 i

10

2.3162

0.338 i

11

-4.6111

0.667

12

1.6999 + 5.3312 i

0.213 - 0.798 i

13

1.6999 - 5.3312 i

0.213 + 0.798 i

14

-40.2633

1.47 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	-2.2575	1.41

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	-0.3276	0.0614
2	-0.855	0.0993 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	-0.7732	0.482
2	1.6974	3.14

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	-0.9385	1.72
2	1.4547	0.844
3	-2.7429	0.833 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	1.2488	0.457
2	-1.2595 + 0.2706 i	1.65 - 1.03 i
3	-1.2595 - 0.2706 i	1.65 + 1.03 i
4	131.6219	4.86 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	-1.1965 + 0.3195 i	1.62 - 0.359 i
2	-1.1965 - 0.3195 i	1.62 + 0.359 i
3	1.2757	0.499
4	-19.4413	3.56e3

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	-0.6639	0.082
2	-0.7987 + 0.0618 i	0.042 - 0.0952 i
3	-0.7987 - 0.0618 i	0.042 + 0.0952 i
4	1.2029	0.341
5	-1.3708	699. 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	-0.8297 + 0.4026 i	0.183 + 0.0479 i
2	-0.8297 - 0.4026 i	0.183 - 0.0479 i
3	1.1669	0.277
4	-1.2035 + 0.4556 i	0.0417 - 0.376 i
5	-1.2035 - 0.4556 i	0.0417 + 0.376 i
6	115.5956	11.7 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	-0.5061 + 0.1173 i	0.00556 + 0.00312 i
2	-0.5061 - 0.1173 i	0.00556 - 0.00312 i
3	-0.5063 + 0.1547 i	0.00305 - 0.00667 i
4	-0.5063 - 0.1547 i	0.00305 + 0.00667 i
5	1.1235	0.182
6	-3.9631	0.511

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	-0.9014 + 0.1456 i	0.356 + 0.0481 i
2	-0.9014 - 0.1456 i	0.356 - 0.0481 i
3	1.2576	0.643
4	-0.1345 + 1.3902 i	0.0492 + 0.122 i
5	-0.1345 - 1.3902 i	0.0492 - 0.122 i
6	-0.4801 + 1.5301 i	0.139 - 0.0342 i
7	-0.4801 - 1.5301 i	0.139 + 0.0342 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	-0.9626 + 0.3154 i	0.526 + 0.791 i
2	-0.9626 - 0.3154 i	0.526 - 0.791 i
3	1.1948 + 0.3122 i	0.16 - 0.235 i
4	1.1948 - 0.3122 i	0.16 + 0.235 i
5	1.7604	0.316
6	-1.8951	1.43
7	-3.0676	9.28 i
8	14.2401	10.9 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	-0.9127 + 0.2611 i	0.29 - 0.233 i
2	-0.9127 - 0.2611 i	0.29 + 0.233 i
3	1.2313 + 0.3763 i	0.223 - 0.0876 i
4	1.2313 - 0.3763 i	0.223 + 0.0876 i
5	-1.1708 + 0.6464 i	0.255 - 0.0129 i
6	-1.1708 - 0.6464 i	0.255 + 0.0129 i
7	-1.4972	1.21
8	2.7567	2.68

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	-0.8854 + 0.2175 i	0.203 + 0.00228 i
2	-0.8854 - 0.2175 i	0.203 - 0.00228 i
3	-1.0938	1.38
4	-1.0844 + 0.5813 i	0.213 + 0.106 i
5	-1.0844 - 0.5813 i	0.213 - 0.106 i
6	1.2304 + 0.3813 i	0.219 - 0.0733 i
7	1.2304 - 0.3813 i	0.219 + 0.0733 i
8	2.7544	2.99
9	-34.7857	0.542 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	-0.9549 + 0.3315 i	0.597 + 0.532 i
2	-0.9549 - 0.3315 i	0.597 - 0.532 i
3	1.2329 + 0.3181 i	0.198 - 0.292 i
4	1.2329 - 0.3181 i	0.198 + 0.292 i
5	-1.5857	1.4
6	2.1365	0.447
7	-2.4589	2.93 i
8	5.16	22.4 i
9	-5.4255 + 3.783 i	1.3 + 0.897 i
10	-5.4255 - 3.783 i	1.3 - 0.897 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	-0.955 + 0.3342 i	0.612 + 0.48 i
2	-0.955 - 0.3342 i	0.612 - 0.48 i
3	1.2313 + 0.323 i	0.201 - 0.269 i
4	1.2313 - 0.323 i	0.201 + 0.269 i
5	-1.5067	1.25
6	2.1916	0.506
7	-3.0299	0.747 i
8	-2.8051 + 3.5101 i	0.411 + 0.406 i
9	-2.8051 - 3.5101 i	0.411 - 0.406 i
10	9.4584	3.76 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	-0.9539 + 0.3289 i	0.545 + 0.568 i
2	-0.9539 - 0.3289 i	0.545 - 0.568 i
3	1.2216 + 0.3448 i	0.206 - 0.17 i
4	1.2216 - 0.3448 i	0.206 + 0.17 i
5	-1.6333	1.44
6	2.2225	0.776
7	0.8593 + 2.5653 i	0.165 - 0.276 i
8	0.8593 - 2.5653 i	0.165 + 0.276 i
9	0.3064 + 2.9198 i	0.243 + 0.204 i
10	0.3064 - 2.9198 i	0.243 - 0.204 i
11	-3.0106	1.45 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	-0.954 + 0.3294 i	0.554 + 0.56 i
2	-0.954 - 0.3294 i	0.554 - 0.56 i
3	1.2172 + 0.3488 i	0.195 - 0.154 i
4	1.2172 - 0.3488 i	0.195 + 0.154 i
5	-1.6222	1.42
6	2.3307	1.17
7	1.3788 + 2.1767 i	0.012 - 0.326 i
8	1.3788 - 2.1767 i	0.012 + 0.326 i
9	-2.8981	1.58 i
10	1.581 + 3.4057 i	0.361 + 0.107 i
11	1.581 - 3.4057 i	0.361 - 0.107 i
12	62.3914	1.91 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	-0.9591 + 0.3285 i	0.695 + 0.626 i
2	-0.9591 - 0.3285 i	0.695 - 0.626 i
3	1.2166 + 0.3354 i	0.127 - 0.214 i
4	1.2166 - 0.3354 i	0.127 + 0.214 i
5	1.1167 + 1.0374 i	0.0617 - 0.0472 i
6	1.1167 - 1.0374 i	0.0617 + 0.0472 i
7	1.1665 + 0.9898 i	0.0485 + 0.059 i
8	1.1665 - 0.9898 i	0.0485 - 0.059 i
9	-1.5281 + 0.3455 i	0.86 - 0.493 i
10	-1.5281 - 0.3455 i	0.86 + 0.493 i
11	-2.1428	0.666
12	3.7492	152.

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	-0.9562 + 0.3284 i	0.604 + 0.604 i
2	-0.9562 - 0.3284 i	0.604 - 0.604 i
3	1.2684	0.276
4	1.2969 + 0.2219 i	0.0502 - 0.441 i
5	1.2969 - 0.2219 i	0.0502 + 0.441 i
6	1.2753 + 0.3482 i	0.878 + 1.01 i
7	1.2753 - 0.3482 i	0.878 - 1.01 i
8	-1.8068 + 0.2518 i	1.56 - 1.26 i
9	-1.8068 - 0.2518 i	1.56 + 1.26 i
10	2.3937	0.715 i
11	-5.0215	0.837
12	2.9629 + 4.1011 i	0.411 + 1.15 i
13	2.9629 - 4.1011 i	0.411 - 1.15 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	-0.956 + 0.3284 i	0.598 + 0.602 i
2	-0.956 - 0.3284 i	0.598 - 0.602 i
3	1.2051 + 0.327 i	0.0953 - 0.274 i
4	1.2051 - 0.327 i	0.0953 + 0.274 i
5	1.4982 + 0.3994 i	0.246 + 0.0779 i
6	1.4982 - 0.3994 i	0.246 - 0.0779 i
7	1.7851	0.5
8	-1.7995 + 0.2582 i	1.54 - 1.17 i
9	-1.7995 - 0.2582 i	1.54 + 1.17 i
10	2.3162	0.338 i
11	-4.6111	0.667
12	1.6999 + 5.3312 i	0.213 - 0.798 i
13	1.6999 - 5.3312 i	0.213 + 0.798 i
14	-40.2633	1.47 i