# Exponential Function

# Defining Differential Equation

The solution to with and , that is the solution to the IVP

- ,

is the *exponential function*

- .

The exponential functions and for .

# Computing The Exponential Function

Updating gives

and after summation, assuming ,

- .

With , we thus have

- .

# Compound Capital

The above formula states that is the compund capital after years of interest at a yearly rate of with a starting capital of .

# Varying the Time Step

We shall see below that as increases the approximation improves and can be made as small as we like. Decreasing the time step in the formula

by increasing , we thus obtain for and the following values for :

- 2
- 2.25
- 2.37
- 2.4414
- 2.4883
- 2.5216
- 2.5465
- 2.5937
- 2.6533
- 2.7048
- 2.7169
- 2.7181
- …

Increasing we can this way compute any number of decimals of the number ,

or more generally of for any .

# Properties of the Exponential Function

Properties of the exponential function can be derived from the defining differential equation

. For example, the basic property of the exponential function

follows by noting that viewed as a a function of , satisfies

- ,

which means that , which proves (\ref{expst}).

The exponential is the solution of the differential equation with .

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