square root of s domain transfer function

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I have a question about calculating the square root of the magnitude of a transfer function. When you take the square root, what is happening? My initial idea of a magnitude is a single value, but in this case, the magnitude changes with respect to frequency. So in effect, are you left with an array of values? As well, what is happening to the phase?

The reason I am asking this is because I am having trouble to evaluate the following equation

$$ \sqrt(||sys||^2+||sys2||^2) $$

where sys and sys2 are transfer functions.