Rayleigh's Theorem

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Rayleigh's Theorem is derived from the equation for Energy

If we assume that the circuit is a Voltage applied over a load then for matters of simplicity we can assume
This leaves us with

This is the same as the dot product so to satisfy the condition for complex numbers it becomes

If we substitute and

Substituting this back into the original equation makes it

The time integral becomes This simplifies the above equation such that

Proving that the energy in the time domain is the same as that in the frequency domain