An exponential function ez 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 ez (where z = x + iy) is an entire transcendental function and is an analytic continuation of the function ex from a real axis into a complex plane by the Euler formula: ez = ex+iy = ex(cos x + i sin y).
The exponential function ez can accommodate all complex values except for 0. The equation ez = 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: ez + 2πi = ez .
The theorem of addition ez1 + z2 = ez1 · ez2 is valid for the function ez.
The derivative of the exponential function ez is coincident with the function (ez)' = ez .
The integral of the function ez is ∫ ezdz = ez + C .
For real numbers of x, the value of the function ex 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.