15.5 Exponential Integrals

The Exponential Integral and related functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, A&S Chapter 5.

Function: expintegral_e1 (z) ¶

The Exponential Integral E1(z) defined as

\[E_1(z) = \int_z^\infty {e^{-t} \over t} dt \]

with \(\left| \arg z \right| < \pi\) . (A&S eqn 5.1.1) and (DLMF 6.2E2)

This can be written in terms of other functions. See expintrep for examples.

Categories: Exponential Integrals · Special functions ·

Function: expintegral_ei (x) ¶

The Exponential Integral Ei(x) defined as

\[Ei(x) = - -\kern-10.5pt\int_{-x}^\infty {e^{-t} \over t} dt = -\kern-10.5pt\int_{-\infty}^x {e^{t} \over t} dt \]

with \(x\) real and \(x > 0\). (A&S eqn 5.1.2) and (DLMF 6.2E5)

This can be written in terms of other functions. See expintrep for examples.

Categories: Exponential Integrals · Special functions ·

Function: expintegral_li (x) ¶

The Exponential Integral li(x) defined as

\[li(x) = -\kern-10.5pt\int_0^x {dt \over \ln t} \]

with \(x\) real and \(x > 1\). (A&S eqn 5.1.3) and (DLMF 6.2E8)

This can be written in terms of other functions. See expintrep for examples.

Categories: Exponential Integrals · Special functions ·

Function: expintegral_e (n,z) ¶

The Exponential Integral En(z) (A&S eqn 5.1.4) defined as

\[E_n(z) = \int_1^\infty {e^{-zt} \over t^n} dt \]

with \({\rm Re}(z) > 1\) and \(n\) a non-negative integer.

For half-integral orders, this can be written in terms of erfc or erf. See expintexpand for examples.

Categories: Exponential Integrals · Special functions ·

Function: expintegral_si (z) ¶

The Exponential Integral Si(z) (A&S eqn 5.2.1 and DLMF 6.2#E9) defined as

\[{\rm Si}(z) = \int_0^z {\sin t \over t} dt \]

This can be written in terms of other functions. See expintrep for examples.

Categories: Exponential Integrals · Special functions ·

Function: expintegral_ci (z) ¶

The Exponential Integral Ci(z) (A&S eqn 5.2.2 and DLMF 6.2#E13) defined as

\[{\rm Ci}(z) = \gamma + \log z + \int_0^z {{\cos t - 1} \over t} dt \]

with \(|\arg z| < \pi\) .

This can be written in terms of other functions. See expintrep for examples.

Categories: Exponential Integrals · Special functions ·

Function: expintegral_shi (z) ¶

The Exponential Integral Shi(z) (A&S eqn 5.2.3 and DLMF 6.2#E15) defined as

\[{\rm Shi}(z) = \int_0^z {\sinh t \over t} dt \]

This can be written in terms of other functions. See expintrep for examples.

Categories: Exponential Integrals · Special functions ·

Function: expintegral_chi (z) ¶

The Exponential Integral Chi(z) (A&S eqn 5.2.4 and DLMF 6.2#E16) defined as

\[{\rm Chi}(z) = \gamma + \log z + \int_0^z {{\cosh t - 1} \over t} dt \]

with \(|\arg z| < \pi\) .

This can be written in terms of other functions. See expintrep for examples.

Categories: Exponential Integrals · Special functions ·

Option variable: expintrep ¶

Default value: false

Change the representation of one of the exponential integrals, expintegral_e(m, z), expintegral_e1, or expintegral_ei to an equivalent form if possible.

Possible values for expintrep are false, gamma_incomplete, expintegral_e1, expintegral_ei, expintegral_li, expintegral_trig, or expintegral_hyp.

false means that the representation is not changed. Other values indicate the representation is to be changed to use the function specified where expintegral_trig means expintegral_si, expintegral_ci; and expintegral_hyp means expintegral_shi or expintegral_chi.

Here are some examples for expintrep set to gamma_incomplete:

(%i1) expintrep:'gamma_incomplete;
(%o1)                          gamma_incomplete
(%i2) expintegral_e1(z);
(%o2)                       gamma_incomplete(0, z)
(%i3) expintegral_ei(z);
(%o3)            log(z) - log(- z) - gamma_incomplete(0, - z)
(%i4) expintegral_li(z);
(%o4)     log(log(z)) - log(- log(z)) - gamma_incomplete(0, - log(z))
(%i5) expintegral_e(n,z);
                                                   n - 1
(%o5)                  gamma_incomplete(1 - n, z) z
(%i6) expintegral_si(z);
(%o6) (%i ((- log(%i z)) + log(- %i z) - gamma_incomplete(0, %i z)
                                              + gamma_incomplete(0, - %i z)))/2
(%i7) expintegral_ci(z);
(%o7) log(z) - (log(%i z) + log(- %i z) + gamma_incomplete(0, %i z)
                                               + gamma_incomplete(0, - %i z))/2
(%i8) expintegral_shi(z);
      log(z) - log(- z) + gamma_incomplete(0, z) - gamma_incomplete(0, - z)
(%o8) ---------------------------------------------------------------------
                                        2
(%i9) expintegral_chi(z);
(%o9) 
      (- log(z)) + log(- z) + gamma_incomplete(0, z) + gamma_incomplete(0, - z)
    - -------------------------------------------------------------------------
                                          2

For expintrep set to expintegral_e1:

(%i1) expintrep:'expintegral_e1;
(%o1)                           expintegral_e1
(%i2) expintegral_ei(z);
(%o2)               log(z) - log(- z) - expintegral_e1(- z)
(%i3) expintegral_li(z);
(%o3)       log(log(z)) - log(- log(z)) - expintegral_e1(- log(z))
(%i4) expintegral_e(n,z);
(%o4)                         expintegral_e(n, z)
(%i5) expintegral_si(z);
(%o5) (%i ((- log(%i z)) - expintegral_e1(%i z) + log(- %i z)
                                                   + expintegral_e1(- %i z)))/2
(%i6) expintegral_ci(z);
(%o6) log(z)
          log(- %i z) (expintegral_e1(%i z) + expintegral_e1(- %i z)) log(%i z)
        - ---------------------------------------------------------------------
                                            2
(%i7) expintegral_shi(z);
          log(z) + expintegral_e1(z) - log(- z) - expintegral_e1(- z)
(%o7)     -----------------------------------------------------------
                                       2
(%i8) expintegral_chi(z);
         (- log(z)) + expintegral_e1(z) + log(- z) + expintegral_e1(- z)
(%o8)  - ---------------------------------------------------------------
                                        2

For expintrep set to expintegral_ei:

(%i1) expintrep:'expintegral_ei;
(%o1)                           expintegral_ei
(%i2) expintegral_e1(z);
                                          1
                         log(- z) - log(- -)
                                          z
(%o2)       (- log(z)) + ------------------- - expintegral_ei(- z)
                                  2
(%i3) expintegral_ei(z);
(%o3)                          expintegral_ei(z)
(%i4) expintegral_li(z);
(%o4)                       expintegral_ei(log(z))
(%i5) expintegral_e(n,z);
(%o5)                         expintegral_e(n, z)
(%i6) expintegral_si(z);
(%o6) (%i (log(%i z) + 2 (expintegral_ei(- %i z) - expintegral_ei(%i z))
                                                            %i          %i
                                        - log(- %i z) + log(--) - log(- --)))/4
                                                            z           z
(%i7) expintegral_ci(z);
(%o7) ((- log(%i z)) + 2 (expintegral_ei(%i z) + expintegral_ei(- %i z))
                                                    %i          %i
                                - log(- %i z) + log(--) + log(- --))/4 + log(z)
                                                    z           z
(%i8) expintegral_shi(z);
(%o8) ((- 2 log(z)) + 2 (expintegral_ei(z) - expintegral_ei(- z)) + log(- z)
                                                                          1
                                                                  - log(- -))/4
                                                                          z
(%i9) expintegral_chi(z);
(%o9) 
                                                                             1
   2 log(z) + 2 (expintegral_ei(z) + expintegral_ei(- z)) - log(- z) + log(- -)
                                                                             z
   ----------------------------------------------------------------------------
                                        4

For expintrep set to expintegral_li:

(%i1) expintrep:'expintegral_li;
(%o1)                           expintegral_li
(%i2) expintegral_e1(z);
                                                                 1
                                                log(- z) - log(- -)
                               - z                               z
(%o2)      (- expintegral_li(%e   )) - log(z) + -------------------
                                                         2
(%i3) expintegral_ei(z);
                                               z
(%o3)                         expintegral_li(%e )
(%i4) expintegral_li(z);
(%o4)                          expintegral_li(z)
(%i5) expintegral_e(n,z);
(%o5)                         expintegral_e(n, z)
(%i6) expintegral_si(z);
                             %i z                     - %e z    %pi signum(z)
        %i (expintegral_li(%e    ) - expintegral_li(%e      ) - -------------)
                                                                      2
(%o6) - ----------------------------------------------------------------------
                                          2
(%i7) expintegral_ci(z);
                        %i z                     - %i z
       expintegral_li(%e    ) + expintegral_li(%e      )
(%o7)  ------------------------------------------------- - signum(z) + 1
                               2
(%i8) expintegral_shi(z);
                                   z                     - z
                  expintegral_li(%e ) - expintegral_li(%e   )
(%o8)             -------------------------------------------
                                       2
(%i9) expintegral_chi(z);
                                   z                     - z
                  expintegral_li(%e ) + expintegral_li(%e   )
(%o9)             -------------------------------------------
                                       2

For expintrep set to expintegral_trig:

(%i1) expintrep:'expintegral_trig;
(%o1)                          expintegral_trig
(%i2) expintegral_e1(z);
(%o2) log(%i z) - %i expintegral_si(%i z) - expintegral_ci(%i z) - log(z)
(%i3) expintegral_ei(z);
(%o3) (- log(%i z)) - %i expintegral_si(%i z) + expintegral_ci(%i z) + log(z)
(%i4) expintegral_li(z);
(%o4) (- log(%i log(z))) - %i expintegral_si(%i log(z))
                                      + expintegral_ci(%i log(z)) + log(log(z))
(%i5) expintegral_e(n,z);
(%o5)                         expintegral_e(n, z)
(%i6) expintegral_si(z);
(%o6)                          expintegral_si(z)
(%i7) expintegral_ci(z);
(%o7)                          expintegral_ci(z)
(%i8) expintegral_shi(z);
(%o8)                      - %i expintegral_si(%i z)
(%i9) expintegral_chi(z);
(%o9)            (- log(%i z)) + expintegral_ci(%i z) + log(z)

For expintrep set to expintegral_hyp:

(%i1) expintrep:'expintegral_hyp;
(%o1)                           expintegral_hyp
(%i2) expintegral_e1(z);
(%o2)               expintegral_shi(z) - expintegral_chi(z)
(%i3) expintegral_ei(z);
(%o3)               expintegral_shi(z) + expintegral_chi(z)
(%i4) expintegral_li(z);
(%o4)          expintegral_shi(log(z)) + expintegral_chi(log(z))
(%i5) expintegral_e(n,z);
(%o5)                         expintegral_e(n, z)
(%i6) expintegral_si(z);
(%o6)                     - %i expintegral_shi(%i z)
(%i7) expintegral_ci(z);
(%o7)           (- log(%i z)) + expintegral_chi(%i z) + log(z)
(%i8) expintegral_shi(z);
(%o8)                         expintegral_shi(z)
(%i9) expintegral_chi(z);
(%o9)                         expintegral_chi(z)

Categories: Exponential Integrals ·

Option variable: expintexpand ¶

Default value: false

Expand expintegral_e(n,z) for half integral values in terms of erfc or erf and for positive integers in terms of expintegral_ei.

(%i1) expintegral_e(1/2,z);
                                            1
(%o1)                         expintegral_e(-, z)
                                            2
(%i2) expintegral_e(1,z);
(%o2)                         expintegral_e(1, z)
(%i3) expintexpand:true;
(%o3)                                true
(%i4) expintegral_e(1/2,z);
                            sqrt(%pi) erfc(sqrt(z))
(%o4)                       -----------------------
                                    sqrt(z)
(%i5) expintegral_e(1,z);
                               1
                         log(- -) - log(- z)
                               z
(%o5)       (- log(z)) - ------------------- - expintegral_ei(- z)
                                  2

Categories: Exponential Integrals ·