# Define a Function

• Nov 12th 2005, 06:21 PM
ThePerfectHacker
Define a Function
I was trying to Generalize the Harmonic Function, define H(n)=1+1/2+1/3+..+1/n
for integral values of n. Now how can H(x) be defined in such as way such as it will be countinous and H(x)=H(n) for integral values of x. This is anagolus with the Gamma function as a generalization for the factoril.
Thus given:
H(x) is countinous for x>1 or x=1.
H(1)=1
H(x+1)=H(x)+1/(x+1)
Find a possible H.
• Nov 13th 2005, 11:51 AM
hpe
Quote:

Originally Posted by ThePerfectHacker
I was trying to Generalize the Harmonic Function, define H(n)=1+1/2+1/3+..+1/n
for integral values of n. Now how can H(x) be defined in such as way such as it will be countinous and H(x)=H(n) for integral values of x. This is anagolus with the Gamma function as a generalization for the factoril.
Thus given:
H(x) is countinous for x>1 or x=1.
H(1)=1
H(x+1)=H(x)+1/(x+1)
Find a possible H.

A linear interpolant (or spline interpolant) will do. To find something special, you probably have to impose extra conditions, such as logarithmic convexity in the case of the Gamma function.
• Nov 22nd 2005, 12:10 AM
lewisje
Something based on the logarithm might work; for example, it is known that lim(H(n)-ln(n),n->infinity) is a constant, called the Euler-Mascheroni constant and denoted by gamma.