# Argument of complex number with natural number k

I have the complex number (1 - E^2) (2 + I 2 Pi k) with k a natural number. How can I get the general formula for the argument? It should be Pi + ArcTan[k Pi].

I tried with:

Assumptions->Element[k,Integers]
Assumptions -> k > 0
Arg[(1 - E^2) (2 + 2 I Pi k)]


but it does not work.

The following gets close:

ComplexExpand[
Arg[(1-E^2) (2+2 I Pi k)],
TargetFunctions->{Re, Im}
]


ArcTan[-2, -2 k π]

• Hi @CarlWoll. What does the comma mean? Sep 5 '17 at 21:56
• @GennaroArguzzi Look it up here. Basically, ArcTan[x, y] is the same as ArcTan[y/x] except that the signs of x and y are used to determine the appropriate quadrant. Sep 5 '17 at 22:03
• @Gennaro, you might meet it a lot of times in future work, so you might as well learn about two-argument arctangent. Sep 6 '17 at 0:02
• @CarlWoll the only strange thing is that real part $\frac{1-e^2}{2+2\pi^2 k^2}<0$ and imaginary part $\frac{\pi k(e^2-1)}{2+2\pi^2 k^2}>0$, thus I expect the result Pi+ArcTan[-2,-2 k Pi]. Sep 6 '17 at 7:13
• @Gennaro, read the last sentence of my previous comment a bit more carefully and slowly. Then evaluate FullSimplify[ComplexExpand[Arg[a + b I] == ArcTan[a, b]], {a, b} ∈ Reals] and look at the result for a full minute. Sep 6 '17 at 20:46

Try this:

Simplify[ComplexExpand[Arg[(1 - E^2)*(2 I Pi k)]], k > 0]

(*  -(\[Pi]/2)  *)


Have fun!