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Alan Turing's paper, Chemical Basis of Morphogenesis, is about how a symmetrical embryonic stage(like a blastula) can create an asymmetric organism or pattern. He creates differential equations to show that small randomness can result in a major disturbance of the symmetry.

The paper says on page 4 that

Since the solutions will normally be real one can also write them in the form $R\Sigma Ae^{bt} $ or $\Sigma R(Ae^{bt}) $ ($R$ means 'real part of')

He is referring to considering the real part of the general solution, which is ignoring the imaginary part. But why can you do that? If you solve a physics problem and get a negative value for the mass, something would be wrong as mass can't be negative. So wouldn't it be the same for complex numbers?

Thanks

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  • $\begingroup$ Looks like Turing means the real part of a sum is the sum of real parts. Hint: Let $a,b,c,d$ be real. What is the real part of $a+ib+c+id$ ? This has nothing to do with a physical interpretation, or with the impossibility that mass becomes negative. $\endgroup$
    – Kurt G.
    Commented Sep 1, 2022 at 13:15
  • $\begingroup$ @KurtG., I apologize. I should have put the quote in context of the paper. I have made the edits, but to reiterate, my question was why we can consider the real part of a complex general solution when it is for a physical scenario $\endgroup$ Commented Sep 1, 2022 at 15:45

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Often in physics we want the real solutions to an equation which, due to linearity, has the following property: if $x$ is an in general complex solution, so are $x^\ast$ and the real-valued linear combinations $\Re x,\,\Im x$ thereof. One can then turn complex solutions into real solutions, and can prove these are all the real solutions. Indeed, this is often an especially simple way to find the real solutions.

To take an example similar to but much simpler than the ones Turing has in mind, consider the harmonic oscillation $\ddot{x}=-\omega^2x$ with $\omega>0$. The most general complex solution is $x=Ae^{i\omega t}+Be^{-i\omega t}$ with arbitrary complex constants $A,\,B$. The most general real solution must be a special case of this; indeed, "reality" (meaning the solution is real-valued) is equivalent to the constraint $B=A^\ast$, making the solution $x=2\Re[Ae^{i\omega t}]$. (Don't worry about the $2$; we can just redefine $A$ to absorb that.) You can similarly rewrite this e.g. as $x=2\Im[iAe^{i\omega t}]$, but that's just the same thing, so as a matter of convention people focus on the $\Re$-using notation.

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