CODEWITHSUNDEEP
pythonmathsympycomplex-numbers
jankos
jankos

Reputation: 916

Assuming positive real part of a complex number in Sympy

I cannot figure it out how to assume positive real part of a complex number in Sympy. Example of an Mathematica code:

a = InverseFourierTransform[ R/(I omega - lambda) +  Conjugate[R]/(I omega - Conjugate[lambda]), omega, t,   FourierParameters -> {1, -1}]

Simplify[a, {Re[lambda] < 0, t > 0}]

Similar Sympy code:

import sympy as sym
sym.init_printing()

ω = sym.symbols('omega', real=True, positive=True) 

R, λ = sym.symbols('R, lambda', complex=True)

t = sym.symbols('t', real=True, positive=True)

α = R/(sym.I*ω-λ)+sym.conjugate(R)/(sym.I*ω-sym.conjugate(λ))

sym.inverse_fourier_transform(α, ω, t)

How could I assume real part of lambda to be positive? If I assume lambda to have positive=True, then sympy assumes imaginary=False.

Any ideas?

Upvotes: 1

Views: 1008

Answers (1)

user6655984
user6655984

Reputation:

Create two real symbols x, y, assume x positive, and let λ be x + I*y. 

import sympy as sym
ω, x, t = sym.symbols('omega x t', positive=True) 
y = sym.symbols('y', real=True)
R = sym.symbols('R')
λ = x + sym.I*y
α = R/(sym.I*ω-λ)+sym.conjugate(R)/(sym.I*ω-sym.conjugate(λ))    
res = sym.inverse_fourier_transform(α, ω, t)

The result is

2*pi*R*exp(2*pi*t*(x + I*y)) + 2*pi*exp(2*pi*t*(x - I*y))*conjugate(R)

You can then return to single-symbol λ with substitution: 

λ = sym.symbols('lambda')
res.subs(x + sym.I*y, λ).conjugate().subs(x + sym.I*y, λ).conjugate()

obtaining 

2*pi*R*exp(2*pi*lambda*t) + 2*pi*exp(2*pi*t*conjugate(lambda))*conjugate(R)

(The trick with two conjugations is needed because subs isn't going to replace x - I*y with conjugate(lambda) otherwise.)

Remarks on assumptions

  • complex=True is redundant; real numbers are included in complex numbers (7 is a complex number), so this has no effect
  • real=True is redundant when positive=True is given

Upvotes: 3

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