# Complex resonant frequency not resonant without imaginary part. So can I still just take real part as solution?

I am working with a matrix on a harmonic oscillator problem and the lowest (absolute) frequency $\omega_0$ where the matrix becomes singular is the resonant frequency.

Now I obtained this frequency by calculating it numerically at let's say it is $\omega_0 = 0.3302 - 0.0121i$, ie. a complex frequency.

My question is what is the actual resonant frequency considering there is no such thing as a complex resonant frequency in real life? I know that in various situations you work with complex numbers for convenience and then take the real part at the end to obtain a solution.

However the imaginary part of $\omega_0$ here is necessary to make the matrix singular, if I just take $\omega_0 = 0.3302$ the matrix is not singular and $w_0$ will not be a resonant frequency.

Or have I got it wrong and is it actually fine to just take the real part (and the imaginary part can be sort of considered to be implied)?

• @Asked: without more context it's hard to say. If the complex numbers were introduced in order to simplify calculations -- exponentials in place of trigonometric functions -- and if the equations are linear, then the real and the imaginary parts are both solutions. But if the imaginary represents losses, the Commented Mar 5, 2016 at 12:18
• ... then it is part of the answer; this happens when the imaginary part represents the absorption in optics. One cannot tell by simply looking at an answer; the problem setup is required. Commented Mar 5, 2016 at 12:20
• The complex numbers were not introduced to simplify calculations. The problem is the resonant frequency of bubbles in water. Commented Mar 5, 2016 at 17:04
• then you already have your answer, and it is complex. Here is a paper that uses the complex angular resonant frequency - I'm more familiar with optical applications than with bubbles, but the complex value simply needs to be decoded into two independent parameters; the imaginary part is usually related to damping, as in the reference paper. Commented Mar 5, 2016 at 20:13
• ncbi.nlm.nih.gov/pmc/articles/PMC3855064; I need new fingertips! Commented Mar 5, 2016 at 21:07

• If you had a pole on the imaginary axis which implies no damping then when you project it onto the real axis you get $0$ which seems incorrect. Did you mean to say poles with zero damping are those on the real axis? Because those ones will not change under projection onto the real axis? Commented Mar 15, 2016 at 10:55
• Yes but then for undamped systems we can only have a damped natural frequency of zero? I would have thought that we could have a frequency at any $\omega \in \mathbb{R}$, the natural resonant frequency with no decrease in oscillations, if there was no damping but you seem to be saying $\omega$ has to be zero in this case? Commented Mar 15, 2016 at 11:21