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Thread: Define a Function

  1. #1
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    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.
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  2. #2
    hpe
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    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.
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  3. #3
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    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.
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