Singularities of Møller-Plesset series: example "a51"

[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 [0, 0, 0]
The most stable singularity is highlighted.

No. z_c c Location in the complex plane

Singularities of quadratic [0, 0, 0] approximant

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Table 2. 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

Singularities of quadratic [1, 0, 0] approximant

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Table 3. 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

-0.3529

0.0936

2

-1.3286

0.182 i

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Table 4. 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

-0.644

0.426
Singularities of quadratic [1, 1, 1] approximant

2

1.8943

6.15

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Table 5. 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

-0.5915

0.283
Singularities of quadratic [2, 1, 1] approximant

2

2.1713 + 1.1015 i

1.07 + 0.746 i

3

2.1713 - 1.1015 i

1.07 - 0.746 i

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Table 6. 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

-0.6155

0.363
Singularities of quadratic [2, 2, 1] approximant

2

1.5784

3.83

3

7.0566

0.767 i

4

-20.9446

1.12 i

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Table 7. 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

-0.6

0.288
Singularities of quadratic [2, 2, 2] approximant

2

0.7114 + 0.0215 i

0.0555 - 0.0539 i

3

0.7114 - 0.0215 i

0.0555 + 0.0539 i

4

1.3706

0.509

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Table 8. 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

-0.605

0.319
Singularities of quadratic [3, 2, 2] approximant

2

0.9674

7.36

3

0.9943

2.44 i

4

3.3588 + 0.2845 i

0.563 - 1.14 i

5

3.3588 - 0.2845 i

0.563 + 1.14 i

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Table 9. 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

-0.6258 + 0.1442 i

0.194 + 0.153 i
Singularities of quadratic [3, 3, 2] approximant

2

-0.6258 - 0.1442 i

0.194 - 0.153 i

3

-0.9332

0.494

4

1.1668

0.387

5

-2.186

1.81 i

6

343.1103

8.78 i

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Table 10. 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

-0.7245

2.75e5
Singularities of quadratic [3, 3, 3] approximant

2

-1.0528

0.261 i

3

-0.7961 + 0.7719 i

0.287 + 0.0731 i

4

-0.7961 - 0.7719 i

0.287 - 0.0731 i

5

1.1758

0.383

6

-2.095

0.369

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Table 11. 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

-0.6792 + 0.1303 i

0.413 + 0.22 i
Singularities of quadratic [4, 3, 3] approximant

2

-0.6792 - 0.1303 i

0.413 - 0.22 i

3

1.0092

0.171

4

1.0308 + 0.1638 i

0.0738 + 0.201 i

5

1.0308 - 0.1638 i

0.0738 - 0.201 i

6

-1.4156 + 0.2959 i

2.2 + 2.29 i

7

-1.4156 - 0.2959 i

2.2 - 2.29 i

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Table 12. 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

-0.6765 + 0.1291 i

0.401 + 0.246 i
Singularities of quadratic [4, 4, 3] approximant

2

-0.6765 - 0.1291 i

0.401 - 0.246 i

3

1.0699 + 0.2196 i

0.0383 + 0.264 i

4

1.0699 - 0.2196 i

0.0383 - 0.264 i

5

1.1387

0.192

6

-1.4391 + 0.1857 i

2.19 + 5.85 i

7

-1.4391 - 0.1857 i

2.19 - 5.85 i

8

2244.0148

32.2 i

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

Table 2. 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

Table 3. 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	-0.3529	0.0936
2	-1.3286	0.182 i

Table 4. 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	-0.644	0.426
2	1.8943	6.15

Table 5. 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	-0.5915	0.283
2	2.1713 + 1.1015 i	1.07 + 0.746 i
3	2.1713 - 1.1015 i	1.07 - 0.746 i

Table 6. 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	-0.6155	0.363
2	1.5784	3.83
3	7.0566	0.767 i
4	-20.9446	1.12 i

Table 7. 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	-0.6	0.288
2	0.7114 + 0.0215 i	0.0555 - 0.0539 i
3	0.7114 - 0.0215 i	0.0555 + 0.0539 i
4	1.3706	0.509

Table 8. 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	-0.605	0.319
2	0.9674	7.36
3	0.9943	2.44 i
4	3.3588 + 0.2845 i	0.563 - 1.14 i
5	3.3588 - 0.2845 i	0.563 + 1.14 i

Table 9. 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	-0.6258 + 0.1442 i	0.194 + 0.153 i
2	-0.6258 - 0.1442 i	0.194 - 0.153 i
3	-0.9332	0.494
4	1.1668	0.387
5	-2.186	1.81 i
6	343.1103	8.78 i

Table 10. 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	-0.7245	2.75e5
2	-1.0528	0.261 i
3	-0.7961 + 0.7719 i	0.287 + 0.0731 i
4	-0.7961 - 0.7719 i	0.287 - 0.0731 i
5	1.1758	0.383
6	-2.095	0.369

Table 11. 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	-0.6792 + 0.1303 i	0.413 + 0.22 i
2	-0.6792 - 0.1303 i	0.413 - 0.22 i
3	1.0092	0.171
4	1.0308 + 0.1638 i	0.0738 + 0.201 i
5	1.0308 - 0.1638 i	0.0738 - 0.201 i
6	-1.4156 + 0.2959 i	2.2 + 2.29 i
7	-1.4156 - 0.2959 i	2.2 - 2.29 i

Table 12. 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	-0.6765 + 0.1291 i	0.401 + 0.246 i
2	-0.6765 - 0.1291 i	0.401 - 0.246 i
3	1.0699 + 0.2196 i	0.0383 + 0.264 i
4	1.0699 - 0.2196 i	0.0383 - 0.264 i
5	1.1387	0.192
6	-1.4391 + 0.1857 i	2.19 + 5.85 i
7	-1.4391 - 0.1857 i	2.19 - 5.85 i
8	2244.0148	32.2 i