Logarithmic Integral as Non-Convergent Series
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Theorem
The logarithmic integral can be defined in terms of a non-convergent series.
That is:
- $\displaystyle \operatorname {li} \left({z}\right) = \sum_{i \mathop = 0}^{+\infty} \frac {i! \, z} {\ln^{i + 1} z} = \frac z {\ln z} \left({\sum_{i \mathop = 0}^{+\infty} \frac {i!} {\ln^i z} }\right)$
Proof
From the definition of the logarithmic integral:
- $\displaystyle \operatorname {li} \left({z}\right) = \int_0^z \frac {\mathrm d t}{\ln t}$
Using Integration by Parts:
| \(\displaystyle \operatorname {li} \left({z}\right)\) | \(=\) | \(\displaystyle \left[{\frac t {\ln t} }\right]_0^z - \int_0^t t \, \mathrm d (\ln^{-1} t)\) | $\quad$ | $\quad$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \frac z {\ln z} + \int_0^z \frac {\mathrm d t} {\ln^2 t}\) | $\quad$ $\dfrac 0 {\ln 0} = 0$ and Derivative of Function to Power of Function | $\quad$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \frac z {\ln z} + \frac z {\ln^2 z} + \int_0^z \frac {2 \, \mathrm d t} {\ln^3 x}\) | $\quad$ Integration by Parts, $\dfrac 0 {\ln^2 0} = 0$, $t \, \mathrm d \left({ln^{-2} t}\right) = -2 \,\ln^{-3} t$ | $\quad$ |
This sequence can be continued indefinitely.
We will consider the nature of the terms outside and inside the integral, after a number $n$ of iterations of integration by parts.
Let $u_n$ be the term outside the integral.
Let $v_n$ be the term inside the integral.
After $n$ iterations of Integration by Parts as above, we have:
- $\displaystyle \operatorname {li} \left({z}\right) = u_n + \int_0^z v_n \, \mathrm d t$
- $\displaystyle u_0 = 0$
- $\displaystyle v_0 = \frac 1 {\ln t}$
It follows that:
| \(\displaystyle \operatorname {li} \left({z}\right)\) | \(=\) | \(\displaystyle u_n + \left[{t \, v_n}\right]_0^z - \int_0^z t \, \mathrm d \left({v_n}\right)\) | $\quad$ | $\quad$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle u_n + \left[{t \, v_n}\right]_0^z + \int_0^z -t \, \mathrm d \left({v_n}\right)\) | $\quad$ | $\quad$ |
This is actually a proof by induction using different language.
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which gives us the recurrence relations:
| \((1):\quad\) | \(\displaystyle u_{n + 1}\) | \(=\) | \(\displaystyle u_n + \left[{t \, v_n}\right]_0^z\) | $\quad$ | $\quad$ | ||||||||
| \((2):\quad\) | \(\displaystyle v_{n + 1}\) | \(=\) | \(\displaystyle -t \cdot \frac {\mathrm d} {\mathrm d t} \left({v_n}\right)\) | $\quad$ | $\quad$ |
- By recurrence on $n$, with the following recurrence hypothesis:
| \((\text{R.H.}):\quad\) | \(\displaystyle v_n\) | \(=\) | \(\displaystyle \frac {n!} {\ln^{n + 1} t}\) | $\quad$ | $\quad$ |
When $n = 0$, we have:
- $\displaystyle v_0 = \frac 1 {\ln t} = \frac {0!} {\ln^{0 + 1} t}$
which verifies the hypothesis.
By supposing true at $n$, we have at $n+1$:
| \(\displaystyle v_{n+1}\) | \(=\) | \(\displaystyle -t \cdot \frac {\mathrm d} {\mathrm d t} \left({v_n}\right)\) | $\quad$ from $(2)$ | $\quad$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle -t \cdot \frac {\mathrm d} {\mathrm d t} \left({\frac {n!} {\ln^{n + 1} t} }\right)\) | $\quad$ from $(\text{R.H.})$ | $\quad$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle -t \cdot n! \cdot -\left({n + 1}\right) \cdot \frac 1 t \cdot \ln^{-\left({n + 1}\right) - 1} t\) | $\quad$ Derivative of Function to Power of Function, Derivative of Natural Logarithm Function | $\quad$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle \frac {\left({n + 1}\right)!} {\ln^{n + 2} t}\) | $\quad$ Definition of Factorial | $\quad$ |
So $(\text{R.H.})$ is verified at $n + 1$ if it is verified at $n$.
So it is proved for every $n \in \N$ (since it is true at $n=0$):
| \((3):\quad\) | \(\displaystyle v_n\) | \(=\) | \(\displaystyle \frac {n!} {\ln^{n+1} t}\) | $\quad$ | $\quad$ |
By taking $(1)$, and inserting $(3)$ in, a new expression for $u_{n + 1}$ in function of $u_n$ (recursive expression):
| \(\displaystyle u_{n + 1}\) | \(=\) | \(\displaystyle u_n + \left[{t \cdot \frac {n!} {\ln^{n + 1} t} }\right]_0^z\) | $\quad$ | $\quad$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle u_n + \frac {z \, n!} {\ln^{n + 1} z} - \frac {0 \cdot n!} {\ln^{n + 1} 0}\) | $\quad$ | $\quad$ | |||||||||
| \(\displaystyle \) | \(=\) | \(\displaystyle u_n + \frac {z \, n!} {\ln^{n + 1} z}\) | $\quad$ | $\quad$ |
That is, we can write by expanding:
| \((4):\quad\) | \(\displaystyle u_{n + 1}\) | \(=\) | \(\displaystyle \sum_{i \mathop = 0}^n \frac {z \, i!} {\ln^{i + 1} z}\) | $\quad$ | $\quad$ |
$\blacksquare$
