Entire wave cosine multiplier ($C_t(z)=\cos (ct\sqrt{z\cdot z})$)

For real wave speed $c$ and $t\ge 0$, define the complex-frequency cosine through the entire series $C_t(z)={\sum }_{j\ge 0}(-1{)}^j(ct{)}^{2j}(z\cdot z{)}^j/(2j)!$. The expression is independent of the square-root branch even though the square root itself need not be entire. Writing $z=a+ib$ gives $|\mathrm{Im}\sqrt{z\cdot z}|\le |b|$, and hence $|C_t(z)|\le e^{|c|t|\mathrm{Im}z|}$. The Paley–Wiener–Schwartz theorem therefore makes its inverse Fourier transform a distribution supported in the radius-$|c|t$ ball. Multiplication by $C_t$ evolves initial displacement with zero initial velocity in the wave equation, giving finite propagation speed for compactly supported initial distributions.