### Abstract

The equation z=2 G(z)-exp G(z)+1 (and similar ones obtained from it by substitutions) appears in connection with a variety of problems ranging from pure mathematics (combinatorics; some first order, nonlinear differential equations) over statistical thermodynamics to renormalization theory. It is therefore of interest to solve this equation for G(z) explicitly. It turns out, after study of the complex structure of the z and G planes, that an explicit integral representation of G(z) can be given, which may be directly used for numerical calculations of high precision.

Original language | English (US) |
---|---|

Pages (from-to) | 563-578 |

Number of pages | 16 |

Journal | Communications in Mathematical Physics |

Volume | 83 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1982 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Physics and Astronomy(all)
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*83*(4), 563-578. https://doi.org/10.1007/BF01208716

**Analytic structure and explicit solution of an important implicit equation.** / Hagedorn, R.; Rafelski, Johann.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 83, no. 4, pp. 563-578. https://doi.org/10.1007/BF01208716

}

TY - JOUR

T1 - Analytic structure and explicit solution of an important implicit equation

AU - Hagedorn, R.

AU - Rafelski, Johann

PY - 1982/12

Y1 - 1982/12

N2 - The equation z=2 G(z)-exp G(z)+1 (and similar ones obtained from it by substitutions) appears in connection with a variety of problems ranging from pure mathematics (combinatorics; some first order, nonlinear differential equations) over statistical thermodynamics to renormalization theory. It is therefore of interest to solve this equation for G(z) explicitly. It turns out, after study of the complex structure of the z and G planes, that an explicit integral representation of G(z) can be given, which may be directly used for numerical calculations of high precision.

AB - The equation z=2 G(z)-exp G(z)+1 (and similar ones obtained from it by substitutions) appears in connection with a variety of problems ranging from pure mathematics (combinatorics; some first order, nonlinear differential equations) over statistical thermodynamics to renormalization theory. It is therefore of interest to solve this equation for G(z) explicitly. It turns out, after study of the complex structure of the z and G planes, that an explicit integral representation of G(z) can be given, which may be directly used for numerical calculations of high precision.

UR - http://www.scopus.com/inward/record.url?scp=33750808941&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33750808941&partnerID=8YFLogxK

U2 - 10.1007/BF01208716

DO - 10.1007/BF01208716

M3 - Article

AN - SCOPUS:33750808941

VL - 83

SP - 563

EP - 578

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 4

ER -