I am reading some references about the Lévy-Khintchine formula. I would like to understand the equivalence. The first reference is Takano's paper:
The family of $p$-dimensional i.d. (infinitely divisible) ch. f.'s coincides with the family of ch. f.'s of the form $e^{\psi}$ with $$ \psi(t)=i a^{\prime} t-\frac{1}{2} t^{\prime} \sigma t+\int_{R_{p}}\left(e^{i t^{\prime }x}-1-\frac{i t^{\prime} x}{1+x^{\prime} x}\right) d \nu, \quad\left(t \in R_{p}\right), $$ where
- $a \in R_{p}$ ( $p$-dimensional Euclidean space of column vect ors),
- $\sigma$ is a non-negative definite matrix of the $p$ th order,
- $\nu$ is a measure defined on the family of Borel sets, $$ \int_{|x|<1}|x|^{2} d \nu<\infty, \quad \int_{|x| \leqq 1} d \nu<\infty, \quad|x|=\sqrt{x^{\prime} x}, $$
On the other hand, the wikipedia gives the formula in the levy process context:
The distribution of a Lévy process is characterized by its characteristic function, which is given by the Lévy-Khintchine formula (general for all infinitely divisibile distributions)
\begin{array}{l} \text { If } X=\left(X_{t}\right)_{t \geq 0} \text { is a Lévy process, then its characteristic function } \varphi_{X}(\theta) \text { is given by } \\ \qquad \varphi_{X}(\theta)(t):=\mathbb{E}\left[e^{i \theta X(t)}\right]=\exp \left(t\left(a i \theta-\frac{1}{2} \sigma^{2} \theta^{2}+\int_{\mathbb{R} \backslash\{0\}}\left(e^{i \theta x}-1-i \theta x \mathbf{I}_{|x|<1}\right) \Pi(d x)\right)\right) \\ \text { where } a \in \mathbb{R}, \sigma \geq 0, \text { and } \Pi \text { is a } \sigma \text {-inite measure called the Lévy measure of } X, \text { satisfying the property } \\ \qquad \int_{\mathbb{R} \backslash\{0\}} \min \left(1, x^{2}\right) \Pi(d x)<\infty . \end{array}
Note that $t$ and $\theta$ are the same in the first and sencond formula, respectively. Denote $u = t = \theta$, I don't understand how the two terms are equal: $$\frac{i u^{\prime} x}{1+x^{\prime} x} = i u x \mathbf{I}_{|x|<1}$$
Some help?