= Entire wave cosine multiplier
{title2=$C_t(z)=\cos(ct\sqrt{z\cdot z})$}
For real wave speed $c$ and $t\ge0$, define the complex-frequency cosine through the entire series $C_t(z)=\sum_{j\ge0}(-1)^j(ct)^{2j}(z\cdot z)^j/(2j)!$. The expression is independent of the square-root branch even though the square root itself need not be entire. Writing $z=a+ib$ gives $|\operatorname{Im}\sqrt{z\cdot z}|\le|b|$, and hence $|C_t(z)|\le e^{|c|t|\operatorname{Im}z|}$. 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>.
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