The family of exponential functions is called the exponential family.

There are many forms of these maps,^[2] many of which are equivalent under a coordinate transformation. For example two of the most common ones are:

• ${\displaystyle E_{c}:z\to e^{z}+c}$ 
• ${\displaystyle E_{\lambda }:z\to \lambda *e^{z}}$ 

The second one can be mapped to the first using the fact that ${\displaystyle \lambda *e^{z}.=e^{z+ln(\lambda )}}$ , so ${\displaystyle E_{\lambda }:z\to e^{z}+ln(\lambda )}$  is the same under the transformation ${\displaystyle z=z+ln(\lambda )}$ . The only difference is that, due to multi-valued properties of exponentiation, there may be a few select cases that can only be found in one version. Similar arguments can be made for many other formulas.