Natural Exponential Function

Natural Exponential Function is a exponential function whose exponent base is [math]\displaystyle{ e }[/math].

• Context:
  • It can be defined by the following power series: [math]\displaystyle{ e^z = 1 + \frac{z}{1!} + \frac{z^2}{2!} + \frac{z^3}{3!} + \dots = \sum_{n=0}^{\infty} \frac{z^n}{n!}. }[/math]
  • It can be defined as the inverse transformation of the natural logarithm.
  • It can be represented as symbol exp instead of the conventionally e, i.e. [math]\displaystyle{ e^x=exp(x) }[/math]
  • In the context of Differential Equations, it appears in the general solution of the first-order differential equation [math]\displaystyle{ y(t)\propto exp(-\int a(t) dt) }[/math] and as complex-valued solution in the solution of a second-order linear differential equation as [math]\displaystyle{ e^{2it}=cos2t+sin2t }[/math]
• Example(s):
  • Gaussian Function.
  • Logistic Function.
  • Fourier Transform.
• Counter-Example(s):
  • Natural Logarithm.
  • Linear Regression
• See: Complex Exponential Function, Exponential Function, Natural Logarithm.

---

References

• http://en.wikipedia.org/wiki/Exponential_function
• (Martin Braun, 1992), Differential Equations and their Applications [1]

---