An exponential function e^{z} is an elementary transcendental function defined for any value of z (real or complex) by

where e is the base of natural logarithms.

The exponential function is a solution of the differential equation ω' = ω and can be presented in the entire open complex plane (i.e., in the entire complex plane except when point z = ∞) as:

The exponential function e^{z} (where z = x + iy) is an entire transcendental function and is an analytic continuation of the function e^{x} from a real axis into a complex plane by the *Euler formula*: e^{z} = e^{x+iy} = e^{x}(cos x + i sin y).

The exponential function e^{z} can accommodate all complex values except for 0. The equation e^{z} = a has an infinite number of solutions for any complex number a ≠ 0. The solution is found by the formula z = ln a + i arg a, where ln a is a logarithmic function, reverse of the exponential.

The exponential function is periodic with a purely imaginary period 2πi: e^{z + 2πi} = e^{z} .

The theorem of addition e^{z1 + z2} = e^{z1} · e^{z2} is valid for the function e^{z}.

The derivative of the exponential function e^{z} is coincident with the function (e^{z})' = e^{z} .

The integral of the function e^{z} is ∫ e^{z}dz = e^{z} + C .

For real numbers of x, the value of the function e^{x} increases faster than any degree x for x → ∞ and for x → −∞ , it tends to zero faster than any other degree 1/x, i.e., for any natural n > 1

Exponential functions are often met in problems of radiation heat transfer. A crucial element in such problems are the integral exponents exp (−t/u) du for flat layers, and for cylindrical geometry.