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 Incoming links: Zero-frequency cancellation of a causal response kernel

Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2017 / Paper 1 / 4 / d / Solution 2026-10-18

For an integrable applied voltage, the convolution theorem gives

I (ω) = [\frac{a _{1} p _{1}}{( γ _{1} + iω ) ^{2} + p _{1}^{2}} + \frac{a _{2} p _{2}}{( γ _{2} + iω ) ^{2} + p _{2}^{2}}] V (ω) .

The integrated electric current is the zero-frequency transform,

Q = I (0)

. Since

V (0)

can be arbitrary, zero total electric charge for every such input is equivalent to

\frac{a _{1} p _{1}}{γ _{1}^{2} + p _{1}^{2}} + \frac{a _{2} p _{2}}{γ _{2}^{2} + p _{2}^{2}} = 0.

For nonzero

p_{1}, p_{2}

, this can be written

\frac{a _{1}}{a _{2}} = - \frac{p _{2} ( γ _{1}^{2} + p _{1}^{2} )}{p _{1} ( γ _{2}^{2} + p _{2}^{2} )} .

Because both amplitudes are positive, nonzero

p_{1}, p_{2}

must have opposite signs. If exactly one

p_{i}

is zero there is no cancellation with positive amplitudes; if both vanish the kernel is identically zero and any amplitudes work. In particular, if both frequency parameters were additionally required to be positive, no such nontrivial system would exist. The PDF requires positivity of

a_{i}, γ_{i}

, but allows real fixed

p_{i}

.

This is zero-frequency cancellation of a causal response kernel. The phrase arbitrary voltage is understood within a class for which the net-charge integrals exist; a permanent DC input need not have finite total charge.

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