HARMNUMB

HARMNUMB (LET)

Library Function

Compute harmonic numbers or generalized harmonic numbers.

The generalized harmonic number is

The case where m = 1 is referred to as the harmonic number and has the formula

The m parameter is restricted to values greater than 1.

LET <y> = HARMNUMB(<n>)             <SUBSET/EXCEPT/FOR qualification>
where <n> is a number, variable, or parameter (> 1);
            <y> is a variable or a parameter (depending on what <x> is) where the computed harmonic number is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax computes the harmonic number.

LET <y> = HARMNUMB(<n>,<m>)
                        <SUBSET/EXCEPT/FOR qualification>
where <n> is a positive integer number, parameter or variable;
            <m> is a number, parmeter, or variable (> 1);
            <y> is a variable or a parameter (depending on what <x> is) where the computed harmonic number is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax computes the generalized harmonic number.

LET A = HARMNUMB(200)
LET A = HARMNUMB(200,1.5)

The zeta function is the limiting case of the generalized harmonic number as n goes to infinity.

The Zipf probability mass and cumulative distribution functions can be defined in terms of generalized harmonic numbers.

None

None

ZETA	= Compute the Riemann zeta function.
ZIPPDF	= Compute the Zipf probability mass function.
ETA	= Compute the eta function.
CATLAN	= Compute the Catlan Beta function.
LAMBDA	= Compute the lambda function.

"Atlas For Computing Mathematical Functions", William J. Thompson, John wiley & Sons, 1997.

"AMS 55: Handbook of Mathematical Functions", Abramowitz and Stegun, Eds., Washington, DC, National Bureau of Standards, 1964.

Special Functions

2006/5

 
multiplot corner coordinates 0 0 100 95
multiplot scale factor 2
case asis
label case asis
title case asis
tic offset units screen
tic offset 3 3
title displacement 2
y1label displacement 17
x1label displacement 12
x1label N
y1label Harmonic Number
.
multiplot 2 2
.
let m = 1.5
title M = ^m
plot harmnumb(n,m) for n = 1 1 100
.
let m = 2.0
title M = ^m
plot harmnumb(n,m) for n = 1 1 100
.
let m = 2.5
title M = ^m
plot harmnumb(n,m) for n = 1 1 100
.
let m = 3.0
title M = ^m
plot harmnumb(n,m) for n = 1 1 100
.
end of multiplot
.
justification center
move 50 97
text Generalized Harmonic Numbers (N = 1 to 100)