Characteristic function of a normal law

Write $g(x)=e^{-x^2\mathrm{/}2}\mathrm{/}\sqrt{2\pi }$ and let $Z$ be the coordinate under its probability law. By [F1]–[F2], this is a probability law with density $g$. For any measurable complex $h$ with $\int \mathrm{\mid }h\mathrm{\mid }g<\mathrm{\infty }$, apply [F11] to the positive and negative parts of $\mathrm{Re}h$ and $\mathrm{Im}h$. The definitions in [F19] then give $\mathbb{E}h(Z)=\int h(x)g(x)dx$; in particular, every density-integral identity used below is covered. The derivative rules give $g\mathrm{\prime }(x)=-xg(x)$. All compact-interval functions below are continuously differentiable, so [F6]–[F9] apply also to each real and imaginary component. AC is used through the normal-law construction and the countable-choice Riemann/Lebesgue bridge; no sequence of arbitrary witnesses is selected.

For $R\ge 1$, the series in [F20] has nonnegative terms at $R^2\mathrm{/}2$, so $e^{R^2\mathrm{/}2}\ge (R^2\mathrm{/}2{)}^2\mathrm{/}2=R^4\mathrm{/}8$. By [F18], $e^{-R^2\mathrm{/}2}=1\mathrm{/}{e}^{R^2\mathrm{/}2}$; hence $0<g(R)\le 8\mathrm{/}(\sqrt{2\pi }{R}^4)$ and both $g(R)$ and $Rg(R)$ tend to zero. For each positive integer $R$, FTC on each half interval gives ${\int }_{-R}^R\mathrm{\mid }x\mathrm{\mid }g(x)dx=2[g(0)-g(R)]$. Thus MCT proves $\mathbb{E}\mathrm{\mid }Z\mathrm{\mid }=2g(0)<\mathrm{\infty }$. Also ${\int }_{-R}^{R}xg(x)dx=g(-R)-g(R)=0$; DCT with majorant $\mathrm{\mid }x\mathrm{\mid }g(x)$ proves $\mathbb{E}Z=0$. Integration by parts with $u=x,v=g$ gives ${\int }_{-R}^R{x}^{2}g(x)dx={\int }_{-R}^{R}g(x)dx-2Rg(R)$. MCT and [F2] now give $\mathbb{E}{Z}^2=1$. All truncations here use the explicit integers R.

By the finite first moment and [F12], ${\phi }_Z\mathrm{\prime }(t)=i\int x{e}^{itx}g(x)dx$. On $[-R,R]$, componentwise integration by parts, using [F14]–[F15], gives $\int x{e}^{itx}g(x)dx=-[e^{itx}g(x){]}_{-R}^R+it\int e^{itx}g(x)dx$. The boundary has modulus at most $2g(R)\to 0$. The two integrands are dominated respectively by $\mathrm{\mid }x\mathrm{\mid }g(x)$ and $g(x)$, already integrable. DCT therefore gives ${\phi }_Z\mathrm{\prime }(t)=-t{\phi }_Z(t)$ for every t, with no improper differentiation left unjustified.

By the real-component product and chain rules, $H(t)=e^{t^2\mathrm{/}2}{\phi }_Z(t)$ has derivative zero. Apply [F16] to its real and imaginary parts on every real interval. Since $H(0)=\int g=1$, $H(t)=1$ for all t, hence ${\phi }_Z(t)=e^{-t^2\mathrm{/}2}$. For $X=m+\sigma Z$ in law, [F17] yields $e^{imt}{\phi }_Z(\sigma t)$, and [F18] combines the exponents. Its moments follow by expanding the finite integrable expressions: $\mathbb{E}X=m+\sigma \mathbb{E}Z=m$ and $\mathbb{E}(X-m{)}^2={\sigma }^2\mathbb{E}{Z}^2={\sigma }^2$. For $\sigma =0$ the variable equals m almost surely and both formulas give the Dirac law directly.