---

C Library Functions  - exp (3)

---

NAME

EXP(3) - [MATHEMATICS] Base-e exponential function

SYNOPSIS

result = exp(x)

         elemental TYPE(kind=KIND) function exp(x)

          TYPE(kind=KIND),intent(in) :: x

CHARACTERISTICS

o	X may be real or complex of any kind.
o	The return value has the same type and kind as X.

DESCRIPTION

EXP(3) returns the value of e (the base of natural logarithms) raised to the power of X.

"e" is also known as Euler’s constant.

So for either a real or complex scalar X, it returns eˆX , where e is the base of the natural logarithm (approximately 2.718281828459045).

For real inputs, EXP returns a real result.

If X is of type complex, its imaginary part is regarded as a value in radians such that (see Euler’s formula):

        exp((re,im)) = exp(re) * cmplx(cos(im),sin(im),kind=kind(cx))

Since EXP(3) is the inverse function of LOG(3) the maximum valid magnitude of the real component of X is LOG(HUGE(X)).

EXP being elemental, when X is an array (real or complex), the function is applied element‐wise, returning an array of the same shape.

    Numerical Considerations

     For very large real X, the result may overflow to infinity in
     finite‐precision arithmetic. For very small (negative) real X ,
     the result approaches zero. Complex inputs with large imaginary
     parts may produce results with significant numerical errors due
     to the trigonometric functions involved.

OPTIONS

o	X : The type shall be real or complex.

RESULT

The value of the result is E**X where E is Euler’s constant.

If X is of type complex, its imaginary part is regarded as a value in radians.

EXAMPLES

Sample program:

    program demo_exp
    implicit none
    integer,parameter :: dp=kind(0.0d0)
    real              :: x, re, im
    complex           :: cx
    real              :: r_array(3), r_array_result(3)
    complex           :: c_array(2), c_array_result(2)
    integer           :: i

       x = 1.0
       write(*,*)"Euler’s constant is approximately",exp(x)

       !! complex values
       ! given
       re=3.0
       im=4.0
       cx=cmplx(re,im)

       ! complex results from complex arguments are Related to Euler’s formula
       write(*,*)’given the complex value ’,cx
       write(*,*)’exp(x) is’,exp(cx)
       write(*,*)’is the same as’,exp(re)*cmplx(cos(im),sin(im),kind=kind(cx))

       ! exp(3) is the inverse function of log(3) so
       ! the real component of the input must be less than or equal to
       write(*,*)’maximum real component’,log(huge(0.0))
       ! or for double precision
       write(*,*)’maximum doubleprecision component’,log(huge(0.0d0))

       ! but since the imaginary component is passed to the cos(3) and sin(3)
       ! functions the imaginary component can be any real value

       ! Real array example
       r_array = [0.0, 1.0, -1.0]
       r_array_result = exp(r_array)
       do i = 1, size(r_array)
         write(*, ’(A, I0, A, F15.10)’) "exp(r_array(", i, ")) = ", r_array_result(i)
       enddo

       ! Complex array example
       c_array = [cmplx(0.0, 0.0, kind=dp), cmplx(1.0, 1.0, kind=dp)]
       c_array_result = exp(c_array)
       do i = 1, size(c_array)
         write(*, ’(A, I0, A, F15.10, A, F15.10, A)’) "exp(c_array(", i, ")) = (", &
         real(c_array_result(i)), ", ", aimag(c_array_result(i)), ")"
       enddo
    end program demo_exp

Results:

     >  Euler’s constant is approximately   2.71828175   
     >  given the complex value              (3.00000000,4.00000000)
     >  exp(x) is           (-13.1287832,-15.2007847)
     >  is the same as           (-13.1287832,-15.2007847)
     >  maximum real component   88.7228394   
     >  maximum doubleprecision component   709.78271289338397    
     > exp(r_array(1)) =    1.0000000000
     > exp(r_array(2)) =    2.7182817459
     > exp(r_array(3)) =    0.3678794503
     > exp(c_array(1)) = (   1.0000000000,    0.0000000000)
     > exp(c_array(2)) = (   1.4686938524,    2.2873551846)

STANDARD

FORTRAN 77

SEE ALSO

o	LOG(3)

RESOURCES

o	Wikipedia:Exponential function
o	Wikipedia:Euler’s formula

Fortran intrinsic descriptions (license: MIT) @urbanjost

---

Nemo Release 3.1

exp (3)

June 29, 2025

Generated by manServer 1.08 from d1c64b66-ae3f-4442-9e9c-7a5f6f4a2f7a using man macros.