In electronics you use frequencies from the complex numbers. These have a real part (the cosine terms) and an imaginary part (the sine terms), so the Taylor expansions do too. But why (use complex frequencies)? When I studied electronics this wasn't really explained, we just did it. That is, what motivation is there, as they say? How to justify this choice of an imaginary sine wave term or sum of terms?
Because in electronics current is not always in phase with voltage. Imagine an AC waveform feeding a resistor. Current and voltage are in phase; both are sine waves and they line up. In impedance terms the impedance is all real. Now imagine it feeding a resistor and an inductor in series. Since it takes some time to build up the current in the inductor, the current will "lag" the voltage waveform. When the voltage is positive, the current will slowly become positive in the inductor; it will rise from zero to some value. When the voltage hits zero, now there's a lot of current in the inductor. Then the voltage reverses and the current slowly starts to reverse in the inductor. But since it takes some time, there is a period of time where voltage is negative and current is positive. Often this is described as the current lagging the voltage. (or the voltage leading the current.) This may explain it better: Please Register or Log in to view the hidden image! To have some way to express this, we use imaginary numbers, which allow you to express both a magnitude (the real part, or the voltage) and the phase difference of another part (the imaginary part, or the phase offset of the current.) This is important in designing complex high frequency networks like filters, antennas and transmission lines.
Ok, this I know about. So you want to represent the phase changes between voltages and currents, in some signal propagating through a circuit. Right, and this phase difference changes as the frequency does; there is a complex impedance. In fact, I know you can describe your inductive circuit with a transfer function, a complex polynomial expression. Because I passed 3rd year electronics. So it's a way to express it? But what's wrong with using real numbers? What is it about a voltage and a current out of phase. Clearly a voltage is different to a current, but they're in the same phase space. So what?
Nothing is wrong with using real numbers. It's easier to perform many operations with imaginary numbers than real ones (either rectangular or polar.)
So I should be able to write a transfer function with real polynomials? For an inductive circuit, say? . . the motivation is it makes the job of designing circuits easier? Still doesn't sound like we're there yet.
Well, the motivation is to make the math easier, which makes it easier to get to the design. Yes. (Don't ask me to do it; that course was a long time ago and nowadays I use tools for that.)
The nicest thing about using complex exponentials is that Euler's identity connects them to sin and cos, and thus complicated geometric identities can be easily derived and summarized using the rules of exponentials, thus greatly simplifying many calculations in signal processing. You can describe any circuit response without any reference to complex numbers, but the computations are uglier that way.
Ok. I had a vague notion occur to me during the course, when we learned how to do the Laplace transform, about a relation between the real and imaginary parts of the transfer function. Roughly, you can ignore either the real part or the imaginary part of the complex response (to real input waveforms), which one depends on the form of the expression after a transform. That is, you want a real response function of frequency but you can get this from the imaginary part of the transform back into the time domain, and sometimes it's easier to do that than futz with the real part. So a complex frequency is actually a simpler object to deal with; besides a field of complex numbers can be a scalar field.
The difference between Fourier and Laplace transforms in circuits is that the Laplace transform solution takes initial conditions into account and gives you the complete circuit response from that moment onwards. The Fourier approach assumes that the circuit has been stimulated by a periodic signal for an effectively infinite amount of time prior to analysis, ignoring the transient response and focusing on long-term asymptotic behaviour. You don't ignore the real or complex part of the transfer function when doing these calculations; rather, you just focus on magnitudes and relative complex phases. Given that the product of two complex numbers has a magnitude equal to the product of the two input magnitudes, and a complex phase equal to the sum of the input complex phases, it's really simple to identify the relationships between phases and amplitudes of sinusoidal inputs and outputs. Like I say you can do all this without complex numbers, but then you have to invoke lots of messy trig identities instead which are by contrast neatly built into and easily derivable from Euler's identity.
Yes, problems involving any sort of periodic phenomenon, because of the Euler representation as Cpt Bork points out. Hence the applications in waves and QM.
From memory the most exciting thing about Fourier Transforms was that the FT of an impulse is the frequency response - with the result that you can hit (say) a bridge with a sledgehammer and see all the resonances (on a spectrum analyser). [analyser is correct UK spelling and I am from the UK]
From the introduction in my notes to the Laplace transform: In this section we complete our formal study of network analysis by extending the ideas of [transfer functions and thevenin and norton equivalents] to calculating the complete response to an arbitrary excitation rather than the steady-state response to an \(e^{st}\) (eternal exponential) input. Ok. Well, I definitely recall doing just that. It was sufficient to have the imaginary response from which you derive the real response. It was often easier to do this, I also recall. Of course, you don't really "ignore" either the real or imaginary parts because both are needed for the Laplace transform into the frequency domain. Ok, simplification is the motivation then.
The Laplace transform doesn't normally have an imaginary component, it's usually applied to real functions of time and gives a real-valued function as output. Going back to Fourier analysis, the idea when describing a circuit's response to a sinusoidal signal is to take either the real or imaginary part of the complex voltage/current as your final answer, but when calculating that voltage/current, you leave everything expressed in terms of complex numbers and Euler's identity in order to find relative phases and amplitudes, otherwise you're back to using messy trig identities. Yes, simplification is the motive. In Quantum mechanics complex numbers are fundamental and irreplaceable, elsewhere they are usually just a convenience. There are lots of math problems and theorems that are much easier to solve and prove with complex analysis techniques than with conventional real number techniques, and in some cases I think complex variable methods are the only known ways to complete many such solutions and proofs.
I have some notes about mathematical physics; boundary value problems, asymptotic expansion etc. Mostly it's formulas and proofs of this and that, so not very annotated. What my old network analysis notes say about the mathematics reads thusly: " Consider a function \( f(t) = Ae^{st}\), where A and s are complex numbers in general. i.e. f is a complex-valued function of t, a real variable. Clearly f cannot be equated to any physical quantity or process, however functions of this form provide a very useful representation of time-dependent physical quantities [such as voltages and currents]. We have \(s = \sigma + j \omega\;\;| j = \sqrt {-1}\); and write \( \vec{A} = |A|e^{j\alpha}\;\;| \alpha = arg\{\vec{A}\}\) So \( f(t) = Ae^{\sigma t}e^{j(\omega t + \alpha)} = Ae^{\sigma t}\{ cos(\omega t + \alpha) +jsin(\omega t + \alpha)\}\) And \( \Re\{f(t)\} = Ae^{\sigma t} cos(\omega t + \alpha) ;\Im\{f(t)\} = Ae^{\sigma t}sin(\omega t + \alpha)\) The real and imaginary parts of f generate the same set of time functions, and so with no loss of generality, we can take the real part to represent physical waveforms."
Your notes must be referring to classical physical processes. There's no way to state the Schrodinger equation or just about any other element of quantum mechanics without incorporating complex numbers and variables, it's more than just a convenience in that case.
