I'd like to take the complex number $(1 + 0i)$ and (i) multiply it by $0.5$ and (ii) add 45 degrees to it (without changing its magnitude). But I'd like to get its complex represenation in non-polar coordinates (i.e., some $(x +yi)$).
How do I do compute like this in Mathematica?
By adding 45 degrees do you mean rotating by 45 degrees?
Let's do this symbolically first. Multiply a complex number x + I y by a scalar a and then rotate by θ.
e1 = a (x + I y) E^(I θ)
a E^(I θ) (x + I y)
Then to get this into the x + I y form you do
ComplexExpand[e1]
a x Cos[θ] - a y Sin[θ] + I (a y Cos[θ] + a x Sin[θ])
In the numerical case this is just
1 0.5 E^(I π/4)
0.353553 + 0.353553 I
1/2*Exp[I*45*Degree] // ComplexExpand
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