1. ## 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.

2. 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.

3. 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.