Keyboard shortcuts

Press ← or → to navigate between chapters

Press S or / to search in the book

Press ? to show this help

Press Esc to hide this help

Special Functions

expintegral_chi (z) — Function

The Exponential Integral Chi(z) (https://personal.math.ubc.ca/~cbm/aands/page_231.htmA&S eqn 5.2.4 and https://dlmf.nist.gov/6.2#E16DLMF 6.2#E16) defined as

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

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

with $|\arg z| < \pi.$

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

See also: expintrep.

expintegral_ci (z) — Function

The Exponential Integral Ci(z) (https://personal.math.ubc.ca/~cbm/aands/page_231.htmA&S eqn 5.2.2 and https://dlmf.nist.gov/6.2#E13DLMF 6.2#E13) defined as

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

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

with $|\arg z| < \pi.$

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

See also: expintrep.

expintegral_e (n, z) — Function

The Exponential Integral En(z) (https://personal.math.ubc.ca/~cbm/aands/page_228.htmA&S eqn 5.1.4) defined as

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

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

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. expintexpand for examples.

See also: erfc, erf, expintexpand.

expintegral_e1 (z) — Function

The Exponential Integral E1(z) defined as

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

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

with $\left| \arg z \right| < \pi.$ (https://personal.math.ubc.ca/~cbm/aands/page_228.htmA&S eqn 5.1.1) and (https://dlmf.nist.gov/6.2E2DLMF 6.2E2)

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

See also: expintrep.

expintegral_ei (x) — Function

The Exponential Integral Ei(x) defined as

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

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

with $x$ real and $x > 0$. (https://personal.math.ubc.ca/~cbm/aands/page_228.htmA&S eqn 5.1.2) and (https://dlmf.nist.gov/6.2E5DLMF 6.2E5)

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

See also: expintrep.

expintegral_li (x) — Function

The Exponential Integral li(x) defined as

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

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

with $x$ real and $x > 1$. (https://personal.math.ubc.ca/~cbm/aands/page_228.htmA&S eqn 5.1.3) and (https://dlmf.nist.gov/6.2E8DLMF 6.2E8)

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

See also: expintrep.

expintegral_shi (z) — Function

The Exponential Integral Shi(z) (https://personal.math.ubc.ca/~cbm/aands/page_231.htmA&S eqn 5.2.3 and https://dlmf.nist.gov/6.2#E15DLMF 6.2#E15) defined as

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

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

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

See also: expintrep.

expintegral_si (z) — Function

The Exponential Integral Si(z) (https://personal.math.ubc.ca/~cbm/aands/page_231.htmA&S eqn 5.2.1 and https://dlmf.nist.gov/6.2#E9DLMF 6.2#E9) defined as

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

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

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

See also: expintrep.

expintexpand — Variable

Default value: false

Expand expintegral_005fe for half integral values in terms of erfc or erf and for positive integers in terms of expintegral_ei.

maxima

(%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

See also: expintegral_e, erfc, erf, expintegral_ei.

expintrep — Variable

Default value: false

Change the representation of one of the exponential integrals, expintegral_005fe, expintegral_005fe1, or expintegral_005fei to an equivalent form if possible.

The possible values for expintrep are:

false — The representation is not changed. gamma_incomplete — The representation uses gamma_incomplete. expintegral_e1 — The representation uses expintegral_e1. expintegral_ei — The representation uses expintegral_ei. expintegral_li — The representation uses expintegral_li. expintegral_trig — The representation uses expintegral_si or expintegral_ci. expintegral_hyp — The representation uses expintegral_shi or expintegral_chi.

Here are some examples for expintrep set to expintrep_002dgamma_002dincomplete:

maxima

(%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);
(%o8) (log(z) - log(- z) + gamma_incomplete(0, z)
                                    - gamma_incomplete(0, - z))/2


(%i9) expintegral_chi(z);
(%o9) - (- log(z) + log(- z) + gamma_incomplete(0, z)
                                    + gamma_incomplete(0, - z))/2

For expintrep set to expintrep_002dexpintegral_002de1:

maxima

(%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);
(%o8) 
    - log(z) + expintegral_e1(z) + log(- z) + expintegral_e1(- z)
  - -------------------------------------------------------------
                                  2

For expintrep set to expintrep_002dexpintegral_002dei:

maxima

(%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)
                                              %i          %i
  - expintegral_ei(%i z)) - log(- %i z) + log(--) - log(- --)))/4
                                              z           z


(%i7) expintegral_ci(z);
(%o7) (- log(%i z) + 2 (expintegral_ei(%i z)
                                               %i          %i
 + expintegral_ei(- %i z)) - log(- %i z) + log(--) + log(- --))/4
                                               z           z
 + log(z)


(%i8) expintegral_shi(z);
(%o8) (- 2 log(z) + 2 (expintegral_ei(z) - expintegral_ei(- z))
                                                            1
                                         + log(- z) - log(- -))/4
                                                            z


(%i9) expintegral_chi(z);
(%o9) (2 log(z) + 2 (expintegral_ei(z) + expintegral_ei(- z))
                                                            1
                                         - log(- z) + log(- -))/4
                                                            z

For expintrep set to expintrep_002dexpintegral_002dli:

maxima

(%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
(%o6) - (%i (expintegral_li(%e    ) - expintegral_li(%e      )
                                                %pi signum(z)
                                              - -------------))/2
                                                      2


(%i7) expintegral_ci(z);
                       %i z                     - %i z
      expintegral_li(%e    ) + expintegral_li(%e      )
(%o7) -------------------------------------------------
                              2
                                                  - signum(z) + 1


(%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 expintrep_002dexpintegral_002dtrig:

maxima

(%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 expintrep_002dexpintegral_002dhyp:

maxima

(%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)

See also: false, expintegral_e, expintegral_e1, expintegral_ei, expintrep, gamma_incomplete, expintegral_li, expintegral_si, expintegral_ci, expintegral_shi, expintegral_chi, expintrep-gamma-incomplete, expintrep-expintegral-e1, expintrep-expintegral-ei, expintrep-expintegral-li, expintrep-expintegral-trig, expintrep-expintegral-hyp.