# expressible in closed form

An expression is , if it can be converted (simplified) into an expression containing only elementary functions^{}, combined by a finite amount of rational operations^{} and compositions.
Thus, such a closed form^{} must not e.g. limit signs, integral signs, sum signs and “…”.

For example,

$$\int \frac{dx}{{x}^{4}+1},$$ |

may be expressed in the closed form

$$\frac{1}{4\sqrt{2}}\mathrm{ln}\frac{{x}^{2}+x\sqrt{2}+1}{{x}^{2}-x\sqrt{2}+1}+\frac{1}{2\sqrt{2}}\mathrm{arctan}\frac{x\sqrt{2}}{1-{x}^{2}}+C$$ |

but for

$$\int \frac{dx}{\sqrt{{x}^{4}+1}}\mathit{d}x,$$ |

there exists no closed form.

In certain contexts, the of the “elementary functions” may be enlarged by allowing in it some other functions, e.g. the error function^{}.

Title | expressible in closed form |
---|---|

Canonical name | ExpressibleInClosedForm |

Date of creation | 2013-03-22 18:29:09 |

Last modified on | 2013-03-22 18:29:09 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 8 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 30A99 |

Classification | msc 26E99 |

Related topic | ClosedForm |

Related topic | IrreducibilityOfBinomialsWithUnityCoefficients |

Related topic | ReductionOfEllipticIntegralsToStandardForm |

Defines | closed form |