The Exponential Map

We know exponentials and logarithms as functions . Here, we shall generalize them in an abstract way to the setting of formal power series over a field.

Exponential and Logarithm in

Suppose is a field, consider the ring of formal power series over : Then the exponential and logarithm are elements defined as follows: where is the ideal of generated by .

Proposition

Suppose and are two elements in , and , then is well-defined and belongs to .

Proof Suppose and . Substitute into , we getNote that has lowest degree term of degree , so the th coefficient of is always a finite sum, therefore is well-defined and belongs to .

Corollary

is well-defined and equals for all . is also well-defined and equals for all .

Proof This is a direct consequence of the above proposition.