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The exponential function ex can be defined in two equivalent ways, as an infinite series:
or as the limit of a sequence:
In these definitions, stands for the factorial of n, and x can be any real number, complex number, element of a Banach algebra (for example, a square matrix), or member of the field of p-adic numbers.
For further explanation of these definitions and a proof of their equivalence, see the article Definitions of the exponential function.
When considered as a function defined on the complex plane, the exponential function retains the important properties
for all z and w.
It is a holomorphic function which is periodic with imaginary period and can be written as
where a and b are real values. This formula connects the exponential function with the trigonometric functions and to the hyperbolic functions. Thus we see that all elementary functions except for the polynomials spring from the exponential function in one way or another.
See also Eulers formula in complex analysis Euler's formula.
Extending the natural logarithm to complex arguments yields a multi-valued function, ln(z). We can then define a more general exponentiation:
for all complex numbers z and w. This is also a multi-valued function. The above stated exponential laws remain true if interpreted properly as statements about multi-valued functions.
The exponential function maps any line in the complex plane to a logarithmic spiral in the complex plane with the center at the origin. This can be seen by noting that the case of a line parallel with the real or imaginary axis maps to a line or circle.
The definition of the exponential function given above can be used verbatim for every Banach algebra, and in particular for square matrices. In this case we have
In addition, the derivative of exp at the point x is that linear map which sends u to u · ex.
In the context of non-commutative Banach algebras, such as algebras of matrices or operators on Banach or Hilbert spaces, the exponential function is often considered as a function of a real argument:
where A is a fixed element of the algebra and t is any real number. This function has the important properties
The "exponential map" sending a Lie algebra to the Lie group that gave rise to it shares the above properties, which explains the terminology. In fact, since R is the Lie algebra of the Lie group of all positive real numbers with multiplication, the ordinary exponential function for real arguments is a special case of the Lie algebra situation. Similarly, since the Lie algebra M(n, R) of all square real matrices belongs to the Lie group of all invertible square matrices, the exponential function for square matrices is a special case of the Lie algebra exponential map.
The term double exponential function can have two meanings:
Compare the super-exponential function, which grows even faster.
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