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Sometimes it is useful to split a complex equation into its real and imaginary parts. Consider the following ode

ode = y'''[x] - k^3*y[x] == I*k*a*((2*x - c)*(y''[x] - k^2*y[x]) + 2*y[x])
bc1 = y'[0] == 0;
bc2 = y''[1] + k^2*y[1]/(1 - c) == 0;

in which y[x] is a complex function with a real independent variable x, $a$ is a real parameter, and $k$ and $c$ are complex parameters. If we write y[x]=yr[x]+I yi[x], k=kr+ I ki, and c=cr+I ci, how can I transfer the system in terms of two real equations odereal and odeimag togeter with their boundary conditions bc1real, bc1imag, bc2real, bc2imag.

For example,

y[x_] := yr[x] + I*yi[x]
bc1 /. y -> y[x]
(*I Derivative[1][yi][0] + Derivative[1][yr][0] == 0*)

Problem: We can see that the real and imaginary parts still are written together in the boundary condition. Note also that I did not need to solve the equation at present. What I want is, for example,

bc1real = Derivative[1][yr][0] == 0
bc1imag = Derivative[1][yi][0] == 0

with a similar split-form for the ode and bc2. Thank you for any suggestions.

Nobody
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1 Answers1

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Splitting complex equations (or expressions) into real and imaginary parts

Uses: I, Re, Im, Assumptions, Alternatives (|), Element, Simplify, Map

I find this comes up most most often with equations. So will illustrate it that way. I just made up the equation below. Key idea is that a and c are assumed real. It's usually the case that I occurs explicitly.

exampleComplexEquation = Expand[(a + I)^4 + 2 a^2 == 30 + 40*c*I];

exampleComplexEquation2 = a + b*I == 3 + 4 I;

Two functions are defined as below. Look up Element to understand acceptable values for arg. These functions are meant to illustrate an approach.

realPart[ex_, arg_] := 
 Simplify[Map[Re, ex], Assumptions -> Element[arg, Reals]]

imagPart[ex_, arg_] := 
 Simplify[Map[Im, ex], Assumptions -> Element[arg, Reals]]

And an example of use. The functions are meant as examples rather than complete solutions.

realEquation = realPart[exampleComplexEquation, a | c]

  a^4 == 29 + 4 a^2

imagEquation = imagPart[exampleEquation, a | c]

  a^3 == a + 10 c

You can also use Solve directly in some cases. The syntax for Element here is a bit different.

Solve[exampleComplexEquation2 && Element[{a, b}, Reals], {a, b}]

  {{a -> 3, b -> 4}}

exampleComplexEquation3 = (a + 2*b*I)^2 == 3 + 4*I;

Solve[exampleComplexEquation3 && Element[{a, b}, Reals], {a, b}]

   {{a -> -2, b -> -(1/2)}, {a -> 2, b -> 1/2}}

Hope this helps lead you down a productive road.

PaulCommentary
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  • your approach is informative but it is cannot deal with a simple ode with an exponential. e.g. extending func definition: realPart[eq_, real_, imag_] := Simplify[Map[Re, eq], Assumptions -> Element[real, Reals] && Element[imag,Complexes]]; imagPart[eq_, real_, imag_] := Simplify[Map[Im, eq], Assumptions -> Element[real, Reals] && Element[imag, Complexes]]; testing with ode = Expand[I*(y''[x] + k^2) + a == 0], realode = realPart[ode, x | a, y | k] gives a == Im[k^2] + Im[(y^\[Prime]\[Prime])[x]]. Well, it is expected to yield a == 2 Re[k] Im[k] + Im[(y^\[Prime]\[Prime])[x]]. – Nobody Jul 07 '20 at 04:01