This graph accompanies the Wikipedia article on indefinite sums. It implements the analytic continuation method of Candelpergher (2017) for visualizing Nørlund's principal solution. For educational use; companion to the article's cited sources. f(x) can be changed live.

N=444

Check for poles, branch points, branch cuts on f(x) in this region. It also must have exponential type less than 2π in the imaginary direction in this region.

h\le x\le h+s\left\{-14\le y\le14\right\}

h=0.1

Step size

s=1

f\left(x\right)=\frac{1}{x}

\sum_{k=1}^{\frac{x}{s}}f\left(sk\right)

t=\frac{x-h}{s}

u\left(x\right)=f\left(sx+h\right)

f_{rac}\left(x\right)=x-\operatorname{floor}\left(x\right)

S\left(t\right)=t-\operatorname{floor}\left(t\right)

H(t)=\left\{t\ge0:1,0\right\}

\operatorname{real}\left(\operatorname{total}\left(u\left(t-\left[0...\operatorname{floor}\left(t\right)\right]\right)\right)\cdot H\left(t\right)+F_{fast}\left(S\left(t\right)-1\right)+u\left(S\left(t\right)\right)H\left(-t\right)-\operatorname{total}\left(u\left(t+\left[1...-\operatorname{floor}\left(t\right)\right]\right)\right)\cdot H\left(-t\right)-C\right)

\operatorname{imag}\left(\operatorname{total}\left(u\left(t-\left[0...\operatorname{floor}\left(t\right)\right]\right)\right)\cdot H\left(t\right)+F_{fast}\left(S\left(t\right)-1\right)+u\left(S\left(t\right)\right)H\left(-t\right)-\operatorname{total}\left(u\left(t+\left[1...-\operatorname{floor}\left(t\right)\right]\right)\right)\cdot H\left(-t\right)-C\right)

You may need to set C manually on edge cases

C_{orig}=K\left(-0.0000000001\right)

C=\left\{\left|\operatorname{real}\left(C_{orig}\right)\right|>10^{4}:K\left(-1.0000000001\right),C_{orig}\right\}

conf

K\left(x\right)=\left(\operatorname{total}\left(u\left(t-\left[0...\operatorname{floor}\left(t\right)\right]\right)\right)\cdot H\left(t\right)+F_{fast}\left(S\left(t\right)-1\right)+u\left(S\left(t\right)\right)H\left(-t\right)-\operatorname{total}\left(u\left(t+\left[1...-\operatorname{floor}\left(t\right)\right]\right)\right)\cdot H\left(-t\right)\right)

C_{old}=\left\{h\ge0:F_{fast}\left(f_{rac}\left(-h\right)-1\right)+u\left(f_{rac}\left(-h\right)\right)-\sum_{n=1}^{-\operatorname{floor}\left(-h\right)}u\left(-h+n\right),h<0:F_{fast}\left(f_{rac}\left(-h\right)-1\right)+\sum_{n=0}^{\operatorname{floor}\left(-h\right)}u\left(-h-n\right)\right\}

\left(\sum_{n=0}^{\operatorname{floor}\left(t\right)}u\left(t-n\right)+F_{fast}\left(f_{rac}\left(t\right)-1\right)-C\right)\left\{-1<\operatorname{real}\left(t\right)\right\}

\left(F_{fast}\left(f_{rac}\left(t\right)-1\right)+u\left(f_{rac}\left(t\right)\right)-\sum_{n=1}^{-\operatorname{floor}\left(t\right)}u\left(t+n\right)-C\right)\left\{0>\operatorname{real}\left(t\right)\right\}

\sum_{n=0}^{\operatorname{floor}\left(t\right)}u\left(t-n\right)\cdot H\left(t\right)+F_{fast}\left(S\left(t\right)-1\right)+u\left(S\left(t\right)\right)H\left(-t\right)-\sum_{n=1}^{-\operatorname{floor}\left(t\right)}u\left(t+n\right)H\left(-t\right)-C

F_{base}\left(x\right)=\int_{1}^{x+1}u\left(o\right)do+\frac{u\left(1\right)-u\left(x+1\right)}{2}+i\int_{0}^{14}\frac{\left(u\left(x+1-io\right)-u\left(1-io\right)\right)-\left(u\left(x+1+io\right)-u\left(1+io\right)\right)}{e^{2\pi o}-1}do

L=\left[0...N-1\right]

x_{1}=\frac{1}{N-1}L-1

y_{1}=F_{base}\left(x_{1}\right)

g\left(x\right)=\max\left(1,\min\left(N-1,\operatorname{floor}\left(\left(x+1\right)\cdot\left(N-1\right)\right)+1\right)\right)

F_{fast}\left(x\right)=y_{1}\left[g\left(x\right)\right]+\frac{\left(y_{1}\left[g\left(x\right)+1\right]-y_{1}\left[g\left(x\right)\right]\right)}{\left(x_{1}\left[g\left(x\right)+1\right]-x_{1}\left[g\left(x\right)\right]\right)}\cdot\left(x-x_{1}\left[g\left(x\right)\right]\right)

p_{ink}=\operatorname{rgb}\left(19,97,97\right)

y_{ellow}=\operatorname{rgb}\left(0,0,57\right)

https://radian628.github.io/unofficial-desmos-wiki/misc/floating-point-numbers/
Desmos isn't very precise anyways, so we truncate the integral as for most functions e^2*pi*t will converge extremely fast regardless for the interval of interest

end of conf (closed by default)