Singularities of Møller-Plesset series: example "a75"

[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

0.069

0.00184
Singularities of quadratic [1, 1, 0] approximant

2

0.0729

0.00189 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

-1.4286

0.323
Singularities of quadratic [1, 1, 1] approximant

2

11.3835

2.34

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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

-0.3793 + 0.4656 i

0.00702 + 0.000941 i
Singularities of quadratic [2, 1, 1] approximant

2

-0.3793 - 0.4656 i

0.00702 - 0.000941 i

3

0.6751

0.00835

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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

-0.9789

0.0818
Singularities of quadratic [2, 2, 1] approximant

2

1.9076 + 1.8554 i

0.237 + 0.074 i

3

1.9076 - 1.8554 i

0.237 - 0.074 i

4

-6.6761

0.33 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

-0.1101

0.0000544
Singularities of quadratic [2, 2, 2] approximant

2

-0.1101

0.0000544 i

3

-0.8482

0.0258

4

1.8738

0.127

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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

-0.9386

0.0621
Singularities of quadratic [3, 2, 2] approximant

2

2.1946 + 1.8639 i

0.121 + 0.0912 i

3

2.1946 - 1.8639 i

0.121 - 0.0912 i

4

1.9943 + 3.922 i

0.188 - 0.0311 i

5

1.9943 - 3.922 i

0.188 + 0.0311 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

0.2561 + 0.e-4 i

0.000171 - 0.000171 i
Singularities of quadratic [3, 3, 2] approximant

2

0.2561 - 0.e-4 i

0.000171 + 0.000171 i

3

-0.925

0.0512

4

3.2146 + 1.064 i

0.917 - 1.1 i

5

3.2146 - 1.064 i

0.917 + 1.1 i

6

-13.7888

0.728 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

-0.9744 + 0.0021 i

12.5 - 6.53 i
Singularities of quadratic [3, 3, 3] approximant

2

-0.9744 - 0.0021 i

12.5 + 6.53 i

3

-1.4677 + 0.1736 i

0.651 + 0.502 i

4

-1.4677 - 0.1736 i

0.651 - 0.502 i

5

2.6101

0.985

6

-5.3355

3.77

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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

-0.8621

0.0286
Singularities of quadratic [4, 3, 3] approximant

2

-1.1321

0.0557 i

3

-1.4244

0.296

4

1.0705 + 2.235 i

0.0602 + 0.0152 i

5

1.0705 - 2.235 i

0.0602 - 0.0152 i

6

2.3259 + 2.0662 i

0.0101 + 0.106 i

7

2.3259 - 2.0662 i

0.0101 - 0.106 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

-0.8615

0.0281
Singularities of quadratic [4, 4, 3] approximant

2

-1.1363

0.0559 i

3

-1.435

0.273

4

1.0954 + 2.156 i

0.0566 + 0.00922 i

5

1.0954 - 2.156 i

0.0566 - 0.00922 i

6

2.1629 + 1.9536 i

0.0177 + 0.0898 i

7

2.1629 - 1.9536 i

0.0177 - 0.0898 i

8

-201657.1219

986. i

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Table 13. 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

-0.8529

0.0268
Singularities of quadratic [4, 4, 4] approximant

2

-1.0622

0.0423 i

3

-1.3543

0.235

4

-1.0717 + 1.4443 i

0.00633 + 0.0425 i

5

-1.0717 - 1.4443 i

0.00633 - 0.0425 i

6

-1.1155 + 1.8574 i

0.0587 - 0.0102 i

7

-1.1155 - 1.8574 i

0.0587 + 0.0102 i

8

2.7369

2.33

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Table 14. 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

-0.8317

0.0251
Singularities of quadratic [5, 4, 4] approximant

2

-0.9099

0.0316 i

3

-1.049

6.01

4

-1.3528

0.107 i

5

-1.8225

0.23

6

3.0456

2.01e3

7

-3.3374

0.108 i

8

0.1266 + 3.5687 i

0.0163 + 0.172 i

9

0.1266 - 3.5687 i

0.0163 - 0.172 i

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Table 15. 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

-0.8258 + 0.0269 i

0.00734 + 0.0119 i
Singularities of quadratic [5, 5, 4] approximant

2

-0.8258 - 0.0269 i

0.00734 - 0.0119 i

3

-0.8926

0.0148

4

-2.1463

146. i

5

2.9681

21.3

6

0.3481 + 3.7761 i

0.122 + 0.201 i

7

0.3481 - 3.7761 i

0.122 - 0.201 i

8

-3.4422 + 1.7955 i

0.164 + 0.0778 i

9

-3.4422 - 1.7955 i

0.164 - 0.0778 i

10

57.6228

2.83 i

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Table 16. 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

-0.8026 + 0.0268 i

0.00261 + 0.00452 i
Singularities of quadratic [5, 5, 5] approximant

2

-0.8026 - 0.0268 i

0.00261 - 0.00452 i

3

-0.8436

0.00519

4

2.5259 + 0.0339 i

0.0792 - 0.0712 i

5

2.5259 - 0.0339 i

0.0792 + 0.0712 i

6

-3.0894

0.265 i

7

-1.2685 + 3.4551 i

0.37 + 0.102 i

8

-1.2685 - 3.4551 i

0.37 - 0.102 i

9

3.863

0.919

10

-12.3298

0.579

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Table 17. 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

-0.797 + 0.024 i

0.00227 + 0.00386 i
Singularities of quadratic [6, 5, 5] approximant

2

-0.797 - 0.024 i

0.00227 - 0.00386 i

3

-0.8361

0.00453

4

-0.9998 + 2.0323 i

0.00148 + 0.0474 i

5

-0.9998 - 2.0323 i

0.00148 - 0.0474 i

6

-1.1524 + 2.4196 i

0.0463 - 0.00731 i

7

-1.1524 - 2.4196 i

0.0463 + 0.00731 i

8

3.0034

43.2

9

0.6193 + 3.3242 i

0.0863 + 0.0821 i

10

0.6193 - 3.3242 i

0.0863 - 0.0821 i

11

-3.4451

0.148 i

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Table 18. 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

-0.7847 + 0.0206 i

0.00133 + 0.00228 i
Singularities of quadratic [6, 6, 5] approximant

2

-0.7847 - 0.0206 i

0.00133 - 0.00228 i

3

-0.8155

0.00264

4

1.9064

0.0184

5

1.9449

0.0192 i

6

-2.0725 + 0.8387 i

0.192 - 0.0469 i

7

-2.0725 - 0.8387 i

0.192 + 0.0469 i

8

-2.6511

0.186 i

9

-1.3744 + 3.4073 i

0.166 + 0.107 i

10

-1.3744 - 3.4073 i

0.166 - 0.107 i

11

4.3878

0.532

12

8.7687

1.65 i

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Table 19. 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

-0.7837 + 0.0203 i

0.00127 + 0.00218 i
Singularities of quadratic [6, 6, 6] approximant

2

-0.7837 - 0.0203 i

0.00127 - 0.00218 i

3

-0.8139

0.00252

4

1.6653

0.00797

5

1.6787

0.00808 i

6

-2.0684 + 0.8295 i

0.19 - 0.058 i

7

-2.0684 - 0.8295 i

0.19 + 0.058 i

8

-2.5948

0.193 i

9

-1.2579 + 3.4582 i

0.179 + 0.0758 i

10

-1.2579 - 3.4582 i

0.179 - 0.0758 i

11

3.9189

0.722

12

10.4397

2.67 i

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Table 20. 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

-0.7801 + 0.0182 i

0.00118 + 0.00194 i
Singularities of quadratic [7, 6, 6] approximant

2

-0.7801 - 0.0182 i

0.00118 - 0.00194 i

3

-0.8102

0.00235

4

-1.4891

0.881 i

5

-1.6157

0.738

6

-2.438

0.371 i

7

2.8344

1.93

8

-0.9584 + 2.7968 i

0.0747 + 0.0458 i

9

-0.9584 - 2.7968 i

0.0747 - 0.0458 i

10

1.6517 + 2.7365 i

0.046 - 0.0834 i

11

1.6517 - 2.7365 i

0.046 + 0.0834 i

12

1.7276 + 4.1286 i

0.05 + 0.0938 i

13

1.7276 - 4.1286 i

0.05 - 0.0938 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	0.069	0.00184
2	0.0729	0.00189 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	-1.4286	0.323
2	11.3835	2.34

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	-0.3793 + 0.4656 i	0.00702 + 0.000941 i
2	-0.3793 - 0.4656 i	0.00702 - 0.000941 i
3	0.6751	0.00835

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	-0.9789	0.0818
2	1.9076 + 1.8554 i	0.237 + 0.074 i
3	1.9076 - 1.8554 i	0.237 - 0.074 i
4	-6.6761	0.33 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	-0.1101	0.0000544
2	-0.1101	0.0000544 i
3	-0.8482	0.0258
4	1.8738	0.127

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	-0.9386	0.0621
2	2.1946 + 1.8639 i	0.121 + 0.0912 i
3	2.1946 - 1.8639 i	0.121 - 0.0912 i
4	1.9943 + 3.922 i	0.188 - 0.0311 i
5	1.9943 - 3.922 i	0.188 + 0.0311 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	0.2561 + 0.e-4 i	0.000171 - 0.000171 i
2	0.2561 - 0.e-4 i	0.000171 + 0.000171 i
3	-0.925	0.0512
4	3.2146 + 1.064 i	0.917 - 1.1 i
5	3.2146 - 1.064 i	0.917 + 1.1 i
6	-13.7888	0.728 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	-0.9744 + 0.0021 i	12.5 - 6.53 i
2	-0.9744 - 0.0021 i	12.5 + 6.53 i
3	-1.4677 + 0.1736 i	0.651 + 0.502 i
4	-1.4677 - 0.1736 i	0.651 - 0.502 i
5	2.6101	0.985
6	-5.3355	3.77

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	-0.8621	0.0286
2	-1.1321	0.0557 i
3	-1.4244	0.296
4	1.0705 + 2.235 i	0.0602 + 0.0152 i
5	1.0705 - 2.235 i	0.0602 - 0.0152 i
6	2.3259 + 2.0662 i	0.0101 + 0.106 i
7	2.3259 - 2.0662 i	0.0101 - 0.106 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	-0.8615	0.0281
2	-1.1363	0.0559 i
3	-1.435	0.273
4	1.0954 + 2.156 i	0.0566 + 0.00922 i
5	1.0954 - 2.156 i	0.0566 - 0.00922 i
6	2.1629 + 1.9536 i	0.0177 + 0.0898 i
7	2.1629 - 1.9536 i	0.0177 - 0.0898 i
8	-201657.1219	986. i

Table 13. 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	-0.8529	0.0268
2	-1.0622	0.0423 i
3	-1.3543	0.235
4	-1.0717 + 1.4443 i	0.00633 + 0.0425 i
5	-1.0717 - 1.4443 i	0.00633 - 0.0425 i
6	-1.1155 + 1.8574 i	0.0587 - 0.0102 i
7	-1.1155 - 1.8574 i	0.0587 + 0.0102 i
8	2.7369	2.33

Table 14. 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	-0.8317	0.0251
2	-0.9099	0.0316 i
3	-1.049	6.01
4	-1.3528	0.107 i
5	-1.8225	0.23
6	3.0456	2.01e3
7	-3.3374	0.108 i
8	0.1266 + 3.5687 i	0.0163 + 0.172 i
9	0.1266 - 3.5687 i	0.0163 - 0.172 i

Table 15. 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	-0.8258 + 0.0269 i	0.00734 + 0.0119 i
2	-0.8258 - 0.0269 i	0.00734 - 0.0119 i
3	-0.8926	0.0148
4	-2.1463	146. i
5	2.9681	21.3
6	0.3481 + 3.7761 i	0.122 + 0.201 i
7	0.3481 - 3.7761 i	0.122 - 0.201 i
8	-3.4422 + 1.7955 i	0.164 + 0.0778 i
9	-3.4422 - 1.7955 i	0.164 - 0.0778 i
10	57.6228	2.83 i

Table 16. 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	-0.8026 + 0.0268 i	0.00261 + 0.00452 i
2	-0.8026 - 0.0268 i	0.00261 - 0.00452 i
3	-0.8436	0.00519
4	2.5259 + 0.0339 i	0.0792 - 0.0712 i
5	2.5259 - 0.0339 i	0.0792 + 0.0712 i
6	-3.0894	0.265 i
7	-1.2685 + 3.4551 i	0.37 + 0.102 i
8	-1.2685 - 3.4551 i	0.37 - 0.102 i
9	3.863	0.919
10	-12.3298	0.579

Table 17. 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	-0.797 + 0.024 i	0.00227 + 0.00386 i
2	-0.797 - 0.024 i	0.00227 - 0.00386 i
3	-0.8361	0.00453
4	-0.9998 + 2.0323 i	0.00148 + 0.0474 i
5	-0.9998 - 2.0323 i	0.00148 - 0.0474 i
6	-1.1524 + 2.4196 i	0.0463 - 0.00731 i
7	-1.1524 - 2.4196 i	0.0463 + 0.00731 i
8	3.0034	43.2
9	0.6193 + 3.3242 i	0.0863 + 0.0821 i
10	0.6193 - 3.3242 i	0.0863 - 0.0821 i
11	-3.4451	0.148 i

Table 18. 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	-0.7847 + 0.0206 i	0.00133 + 0.00228 i
2	-0.7847 - 0.0206 i	0.00133 - 0.00228 i
3	-0.8155	0.00264
4	1.9064	0.0184
5	1.9449	0.0192 i
6	-2.0725 + 0.8387 i	0.192 - 0.0469 i
7	-2.0725 - 0.8387 i	0.192 + 0.0469 i
8	-2.6511	0.186 i
9	-1.3744 + 3.4073 i	0.166 + 0.107 i
10	-1.3744 - 3.4073 i	0.166 - 0.107 i
11	4.3878	0.532
12	8.7687	1.65 i

Table 19. 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	-0.7837 + 0.0203 i	0.00127 + 0.00218 i
2	-0.7837 - 0.0203 i	0.00127 - 0.00218 i
3	-0.8139	0.00252
4	1.6653	0.00797
5	1.6787	0.00808 i
6	-2.0684 + 0.8295 i	0.19 - 0.058 i
7	-2.0684 - 0.8295 i	0.19 + 0.058 i
8	-2.5948	0.193 i
9	-1.2579 + 3.4582 i	0.179 + 0.0758 i
10	-1.2579 - 3.4582 i	0.179 - 0.0758 i
11	3.9189	0.722
12	10.4397	2.67 i

Table 20. 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	-0.7801 + 0.0182 i	0.00118 + 0.00194 i
2	-0.7801 - 0.0182 i	0.00118 - 0.00194 i
3	-0.8102	0.00235
4	-1.4891	0.881 i
5	-1.6157	0.738
6	-2.438	0.371 i
7	2.8344	1.93
8	-0.9584 + 2.7968 i	0.0747 + 0.0458 i
9	-0.9584 - 2.7968 i	0.0747 - 0.0458 i
10	1.6517 + 2.7365 i	0.046 - 0.0834 i
11	1.6517 - 2.7365 i	0.046 + 0.0834 i
12	1.7276 + 4.1286 i	0.05 + 0.0938 i
13	1.7276 - 4.1286 i	0.05 - 0.0938 i