According to Wolfram Alpha, harmonic series $H_x$ has the following representation:
$$H_x=\int_{0}^{1}\frac{-1+t^x}{-1+t}dt=\int_{0}^{\infty}\frac{1-e^{-xt}}{-1+e^t}dt,~Re(x)>-1$$
The corresponding graph for $H_x$ will then be
However, Wolfram Alpha also gives the graph for $Re(x)<1$
But either one of the two functions above is undefined for $Re(x)\leq-1$. According to which function does Wolfram Alpha graph the harmonic series $H_x$, for $Re(x)<-1$? Is here such a function that satisfies both the values for $ Re(x)>-1$ and the values graphed by Wolfram Alpha when $Re(x)<-1$?
Best Answer
Harmonic Numbers can also be extended to all $z\in\mathbb{C}$, except the negative integers, as $$H(z)=\sum_{k=1}^\infty\left(\frac1k-\frac1{k+z}\right)$$