Exponential function

Key properties

• Algebraic identity: exp(x+y)=exp(x)·exp(y), which makes it a homomorphism from the additive reals to the positive multiplicative reals.
• Derivative and integral: d/dx exp(x)=exp(x) and ∫exp(x) dx = exp(x)+C, so it is the unique nontrivial function equal to its own derivative up to constant factors.
• Range and monotonicity: For real x, exp(x)>0 for all x and the function is strictly increasing with range (0,∞).
• Inverse: The natural logarithm ln(x) is the inverse of exp(x).
• Complex extension: Via its power series exp(z)=∑_{n=0}^∞ z^n/n!, the function extends to complex arguments and satisfies Euler's formula exp(iθ)=cosθ + i sinθ.

Definition and relations

One standard definition uses the power series exp(x)=∑_{n=0}^∞ x^n/n!, convergent for every real or complex x. Equivalently, e can be defined as exp(1) or as a limit arising from compound growth. For other bases b>0, exponentiation b^x is related by b^x = exp(x ln b). See also the constant e and the fact that e is an irrational number approximately equal to 2.71828.

History and development

The exponential and logarithmic concepts emerged from early studies of compound interest and logarithms. Over time, mathematicians formalized e and the exponential series; Leonhard Euler popularized the notation and many properties in the 18th century. The power-series definition and complex extension were central developments in analysis.

Applications and examples

The exponential function models continuous growth and decay such as population increase, radioactive decay, and continuously compounded interest. It is central in solving linear differential equations, appears in probability theory (for example, the exponential distribution), and underpins transforms used in engineering and physics. In linear algebra and systems theory, the matrix exponential describes solutions of linear systems of differential equations.

Notable facts and distinctions

As x→∞, exp(x) grows faster than any polynomial in x. It is also used to build other transcendental functions and connects algebra, analysis, and geometry through identities like exp(x+2πi)=exp(x). Its self-derivative property makes it a convenient basis for many theoretical and practical constructions.

For further reading and technical references, consult introductory texts on calculus, complex analysis, and differential equations; online resources provide accessible explanations and worked examples.