= Integrated electromotive response of a finite cyclonic event
{title2=$\mathcal E_x=\pi B_0a^4(x\cos x-\sin x)/x^2$}
For the <ideal magnetic response to a cylindrical cyclonic event>, the plane-integrated $x$-component of $\mathbf u\times\mathbf B$ at duration $t$ is
$$
\mathcal E_x(t)=\pi B_0\int_0^\infty\left[t r^2(fg'-gf')\cos(ft)-2rg\sin(ft)\right]dr.
$$
This follows from $(\mathbf u\times\mathbf B)_x=(rfB_z+gA_r)\cos\phi-(g/r)A_\phi\sin\phi$ and azimuthal integration. If boundary terms vanish, integration by parts gives the equivalent form $\pi B_0\int_0^\infty r^2g'[\sin(ft)+tf\cos(ft)]dr$. For $f=g=a^2-r^2$ on $r<a$, zero outside, the response is
$$
\mathcal E_x(t)=\pi B_0a^4\frac{x\cos x-\sin x}{x^2},\qquad x=a^2t.
$$
The value at zero is its continuous limit zero. It is initially $-\pi B_0a^6t/3$, but changes sign repeatedly at the nonzero roots of $\tan x=x$, with oscillatory envelope of order $t^{-1}$. The flow's <kinetic helicity density> is $2f^2\geq0$, showing that the sign of a finite-duration ideal response is not fixed by the instantaneous helicity sign.
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