# Functions That Do Not Have Unique Values

When you ask for the square root of a number , you are effectively asking for the solution to the equation . This equation, however, in general has two different solutions. Both and are, for example, solutions to the equation . When you evaluate the "function" , however, you usually want to get a single number, and so you have to choose one of these two solutions. A standard choice is that should be positive for . This is what the Wolfram Language function Sqrt[x] does.

The need to make one choice from two solutions means that Sqrt[x] cannot be a true *inverse function* for . Taking a number, squaring it, and then taking the square root can give you a different number than you started with.

In[1]:= |

Out[1]= |

In[2]:= |

Out[2]= |

When you evaluate , there are again two possible answers: and . In this case, however, it is less clear which one to choose.

There is in fact no way to choose so that it is continuous for all complex values of . There has to be a "branch cut"—a line in the complex plane across which the function is discontinuous. The Wolfram Language adopts the usual convention of taking the branch cut for to be along the negative real axis.

In[3]:= |

Out[3]= |

In[4]:= |

Out[4]= |

In[5]:= |

Out[5]= |

In[6]:= |

Out[6]= |

When you find an root using , there are, in principle, possible results. To get a single value, you have to choose a particular *principal root*. There is absolutely no guarantee that taking the root of an power will leave you with the same number.

In[7]:= |

Out[7]= |

In[8]:= |

Out[8]= |

There are many mathematical functions which, like roots, essentially give solutions to equations. The logarithm function and the inverse trigonometric functions are examples. In almost all cases, there are many possible solutions to the equations. Unique "principal" values nevertheless have to be chosen for the functions. The choices cannot be made continuous over the whole complex plane. Instead, lines of discontinuity, or branch cuts, must occur. The positions of these branch cuts are often quite arbitrary. The Wolfram Language makes the most standard mathematical choices for them.

Sqrt[z] and z^s | for Re , for Re ( not an integer) |

Exp[z] | none |

Log[z] | |

trigonometric functions | none |

ArcSin[z] and ArcCos[z] | and |

ArcTan[z] | and |

ArcCsc[z] and ArcSec[z] | |

ArcCot[z] | |

hyperbolic functions | none |

ArcSinh[z] | and |

ArcCosh[z] | |

ArcTanh[z] | and |

ArcCsch[z] | |

ArcSech[z] | and |

ArcCoth[z] |

Some branch‐cut discontinuities in the complex plane.

In[9]:= |

Out[9]= |

In[10]:= |

Out[10]= |

In[11]:= |

Out[11]= |