# Category:Exponential Function

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This category contains results about **the exponential function**.

Definitions specific to this category can be found in Definitions/Exponential Function.

The **exponential function** can be defined as the unique particular solution $y = \map f z$ to the first order ODE:

- $\dfrac {\d y} {\d z} = y$

satisfying the initial condition $\map f 0 = 1$.

That is, the defining property of $\exp$ is that it is its own derivative.

## Subcategories

This category has the following 30 subcategories, out of 30 total.

### D

### E

- Examples of Exponential Function (empty)
- Exponential Curves (empty)
- Exponential of Product (4 P)
- Exponential of Sum (16 P)
- Exponential of Zero (7 P)
- Exponential Series (1 P)

### G

- General Exponential Function (empty)

### L

### P

### S

- Sigmoid Function (2 P)

## Pages in category "Exponential Function"

The following 81 pages are in this category, out of 81 total.

### C

### E

- Equivalence of Definitions of Complex Exponential Function
- Equivalence of Definitions of Exponential Function
- Equivalence of Definitions of Real Exponential Function
- Euler's Formula
- Euler's Formula/Real Domain
- Euler's Identities
- Euler's Number to Rational Power permits Unique Continuous Extension
- Exponent Combination Laws/Quotient of Powers
- Exponential Dominates Polynomial
- Exponential Function is Continuous
- Exponential Function is Differentiable
- Exponential Function is Well-Defined/Real
- Exponential Functions are Transcendental
- Exponential is Strictly Convex
- Exponential is Strictly Increasing
- Exponential of Complex Number plus 2 pi i
- Exponential of Natural Logarithm
- Exponential of One
- Exponential of Product
- Exponential of Rational Number is Irrational
- Exponential of Real Number is Strictly Positive
- Exponential of Sum
- Exponential of x not less than 1+x
- Exponential of Zero
- Exponential of Zero and One
- Exponential on Complex Plane is Group Homomorphism
- Exponential on Real Numbers is Group Isomorphism
- Exponential on Real Numbers is Injection
- Exponential Sequence is Eventually Increasing
- Exponential Sequence is Eventually Strictly Positive
- Exponential Sequence is Uniformly Convergent on Compact Sets
- Exponential Tends to Zero and Infinity

### L

### M

### P

- Period of Complex Exponential Function
- Periodicity of Complex Exponential Function
- Power Series Expansion for Exponential Function
- Power Series Expansion for Exponential of Cosine of x
- Power Series Expansion for Exponential of Sine of x
- Power Series Expansion for Exponential of Tangent of x
- Power Series Expansion for Exponential of x by Cosine of x
- Power Series Expansion for Exponential of x by Sine of x
- Power Series Expansion for General Exponential Function
- Power Series Expansion for Integer Power of Exponential Function minus 1
- Primitive of Exponential Function
- Properties of Complex Exponential Function
- Properties of Exponential Function

### R

### S

- Sum of Complex Exponentials of i times Arithmetic Sequence of Angles
- Sum of Complex Exponentials of i times Arithmetic Sequence of Angles/Formulation 1
- Sum of Complex Exponentials of i times Arithmetic Sequence of Angles/Formulation 2
- Sum of Complex Exponentials of i times Arithmetic Sequence of Angles/Formulation 3
- Sum of Exponential of i k x
- Sum of Hyperbolic Sine and Cosine equals Exponential
- Surjective Restriction of Real Exponential Function