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Dataplot Vol 2 Vol 1

EN

Name:
    EN (LET)
Type:
    Library Function
Purpose:
    Compute the Euler number or the Euler polynomial.
Description:
    The Euler numbers can be defined by the recurrence relation:

      E(2n) = -SUM[(2*n)!/((2*k)!(2*n-2*k)!)*E(2*k)   (n>=1),
 the summation is from k = 0 to n-1

    where n defines the order of the Euler number and the factorial function, and E(0) = 1. Note that Euler numbers are only defined for 0 and positive even integers.

    The Euler polynomials can be defined in terms of the Euler numbers:

      E(x,n) = SUM[n!E(k)/(2**k*k!*(n-k)!)*(x-0.5)**(n-k)] 
 where the summation is from k = 0 to n

    where E(k) is the Euler number of order k.

    Dataplot computes this function using the EULERB and EULERP routines from "Computation of Special Functions" (see the Reference section below).

Syntax 1:
    LET <y> = EN(<x>,<n>)             <SUBSET/EXCEPT/FOR qualification>
    where <x> is a number, variable or parameter;
                <n> is a non-negagive integer number, variable or parameter;
                <y> is a variable or a parameter (depending on what <x> and <n> are) where the computed Euler polynomial values are stored;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax computes the Euler polynomial of order .

Syntax 2:
    LET <y> = EN(<n>)             <SUBSET/EXCEPT/FOR qualification>
    where <n> is a non-negagive integer number, variable or parameter;
                <y> is a variable or a parameter (depending on what <n> is) where the computed Euler numbers are stored;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax computes the Euler number of order .

Examples:
    LET A = EN(12)
    LET A = EN(B)
    LET A = EN(2.5,4)
    LET Y = EN(X,N)
Default:
    None
Synonyms:
    None
Related Commands:
    BN = Compute Bernoulli number or polynomial.
    EULER NUMBERS = Generate a sequence of Euler numbers.
Reference:
    "Computation of Special Functions", Shanjie Zhang and Jianming Jin, John Wiley and Sons, 1996, chapter 1.
Applications:
    Probability
Implementation Date:
    1997/12
Program:
    TITLE AUTOMATIC
    PLOT EN(X,2) FOR X = 0 0.01 5

    plot generated by sample program

Date created: 6/5/2001
Last updated: 4/4/2003
Please email comments on this WWW page to alan.heckert@nist.gov.