How i can exp(I*x) --> cos(x)+I*sin(x) ?
I wrote the following for personnal use :
def ThisIsEulerFormula(t):
"""
Return t, but in a simplified form.
We force Sage to see that exp(ix) is equal to cos(x)+i*sin(x) and
then to simplify the trigonometric expressions.
INPUT :
- ``t`` - a complex number
OUTPUT:
a complex number that is equal to t.
EXAMPLE::
sage: var('x')
x
sage: a=exp(I*x)
sage: a.simplify_trig()
e^(I*x)
sage: ThisIsEulerFormula(a)
I*sin(x) + cos(x)
This is useful for detecting that an expression is real::
sage: var('x')
x
sage: c=exp(I*x)+exp(-I*x)
sage: c.simplify_trig()
(e^(2*I*x) + 1)*e^(-I*x)
sage: ThisIsEulerFormula(c)
2*cos(x)
NOTE:
I think that the point is that Sage does not know that x is real.
"""
return (t.real_part()+I*t.imag_part()).simplify_trig()
Hope that helps
Laurent
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