How can I obtain the real or the imaginary part of a complex exponential function in Python

Question

I am using Jupyter in particular. e.g from the equation e^2+j, how can I seperate it in it's real part (e²) and into it's imaginary (e^j)? I have tried: exp(complex(2,1)).real However the error that follows it is: 'Mul' object has no attribute 'real'. An other solution might be to implement the eulers formula to seperate it as cos(2)+j­·sin(1) but no success so far. Generally the problem is when complex numbers appear in power position rather than in usual format (2+j). If someone has any idea upon this matter would be greatly appreciated!

Important edit that fits more in my problem:

In my case the complex equation that i have, came through the dsolve() for a 2nd order differential equation. For the exp() element that exist in nympy this is a valid solution, it's not the same as an arbitary equation. However my equation is just a the above one regarding it's complexity

I include my code:

import scipy as sp
from sympy import*
import sympy as syp
from scipy.integrate import odeint

t, z, w, C2=symbols('t, z, w, C2')
x=Function('x')
eq=x(t).diff(t,2)+2*z*w*x(t).diff(t,1)+w**2*x(t)
sol=dsolve(eq,x(t),ics={x(0):0,x(t).diff(t,1).subs(t,0):2*C2*w*sqrt(z**2-1)})

next I want to substitute the z,w parameters in order to fit my data and then with a loop to make an array that takes the numerical solutions in order to plot them. I've tried the following:

for i in range(1000):
    step.append(i)
numdata=[]
for i in range(1000):
    numdata.append(N(sol.rhs.subs(t,i).subs(w,10).subs(z,0.001)))

However this can't work as the sol is an complex function. After this long journey I try to find (meaning of my life) ways to seperate the real and imaginary parts of this kind of function. Thank you for being with me so far, you are a hero reagrdless of the result.

Answer 1

sympy has re and im functions:

In [113]: exp(complex(2,1))                                                                          
Out[113]: 
                  1.0⋅ⅈ
7.38905609893065⋅ℯ     

In [114]: re(exp(complex(2,1)))                                                                      
Out[114]: 3.99232404844127

In [115]: im(exp(complex(2,1)))                                                                      
Out[115]: 6.21767631236797

In [116]: exp(complex(2,1)).evalf()                                                                  
Out[116]: 3.99232404844127 + 6.21767631236797⋅ⅈ

.real and .imag are attributes (possibly implemented as properties) of numpy arrays (and complex python numbers).

Exploring sympy a bit more:

In [152]: expand(exp(y),complex=True)                                                                
Out[152]: 
   re(y)               re(y)           
ⅈ⋅ℯ     ⋅sin(im(y)) + ℯ     ⋅cos(im(y))

In [153]: expand(exp(complex(2,1)),complex=True)                                                     
Out[153]: 3.99232404844127 + 6.21767631236797⋅ⅈ

---

Your sol:

In [157]: sol                                                                                        
Out[157]: 
                 ⎛        ________⎞           ⎛        ________⎞
                 ⎜       ╱  2     ⎟           ⎜       ╱  2     ⎟
             t⋅w⋅⎝-z - ╲╱  z  - 1 ⎠       t⋅w⋅⎝-z + ╲╱  z  - 1 ⎠
x(t) = - C₂⋅ℯ                       + C₂⋅ℯ                      

In [181]: f1 = sol.rhs.subs({w:10, z:0.001,C2:1})                                                    

In [182]: f1                                                                                         
Out[182]: 
   10⋅t⋅(-0.001 - 0.999999499999875⋅ⅈ)    10⋅t⋅(-0.001 + 0.999999499999875⋅ⅈ)
- ℯ                                    + ℯ                                   

Making a numpy compatible function:

In [187]: f = lambdify(t, f1)                                                                        

In [188]: print(f.__doc__)                                                                           
Created with lambdify. Signature:

func(t)

Expression:

-exp(10*t*(-0.001 - 0.999999499999875*I)) + exp(10*t*(-0.001 +...

Source code:

def _lambdifygenerated(t):
    return (-exp(10*t*(-0.001 - 0.999999499999875*1j)) + exp(10*t*(-0.001 + 0.999999499999875*1j)))


Imported modules:

evaluate it at a range of values:

In [189]: f(np.arange(10))                                                                           
Out[189]: 
array([0.+0.j        , 0.-1.07720771j, 0.+1.78972745j, 0.-1.91766624j,
       0.+1.43181934j, 0.-0.49920326j, 0.-0.57406585j, 0.+1.44310044j,
       0.-1.83494157j, 0.+1.63413971j])

same values with just sympy:

In [199]: [im(f1.evalf(subs={t:i})) for i in range(10)]                                              
Out[199]: 
[0, -1.0772077135423, 1.78972744700845, -1.9176662437755, 1.43181934232583, -0.499203257243971, -0.
574065847629935, 1.44310044143674, -1.83494157235822, 1.63413971490123]

Answer 2

I think this issues arises because you are trying to do a from numpy import * along with from sympy import *

Since both numpy and and sympy have their own definition of exp. The error is telling you that the Mul object does not have a exp method, since the interpreter is now confused between the sympy and numpy methods.

Therefore I would recommend doing this instead -

import numpy as np
import sympy as sp

Reference here

After that you can simply do -

np.exp(complex(2,1)).imag
#Output - 6.217676312367968

np.exp(complex(2,1)).real
#Output - 3.992324048441272

np.exp(complex(2,1))
#Output - (3.992324048441272+6.217676312367968j)

---

EDIT: Since you get the output from a sympy dsolve() you can try using the alternate form of

e^(a+ib) = e^acos(b) + ie^asin(b)

c = complex(2,1)

complex(sp.exp(c.real)*sp.cos(c.imag), sp.exp(c.real)*sp.sin(c.imag))
#Output - (3.992324048441272+6.217676312367968j)

sp.exp(c.real)*sp.cos(c.imag)
#Output - 3.992324048441272

sp.exp(c.real)*sp.sin(c.imag)
#Output - 6.217676312367968

---

EDIT 2 : You could lambdify your function and then solve to get real and imag parts.

expp = lambdify([(t,z,w,C2)],sol.rhs)
expp((1,complex(4,3),4,6))
#output - (4.234414847842685+1.053014400461299j)

expp((1,complex(4,3),4,6)).real
#output - 4.234414847842685

expp((1,complex(4,3),4,6)).imag
#output - 1.053014400461299