ArcTan really bothers me, so is there a way to make Mathematica write Logarithm instead of ArcTan? What I'm after is some kind of a ArcTan to Log converter such as the following list from Wiki

• Try TrigToExp, e.g. TrigToExp @ ArcTan[x] works as you seem to expect Commented Jun 5, 2013 at 16:05
• That's awesome, thanks a lot man. You should post this as an answer. Commented Jun 5, 2013 at 16:06
• So now I'm curious. What's bothering you in ArcTans? Commented Jun 5, 2013 at 18:03
• @Vries They are way too restricted for me, $\log$s are much more general and easier to work with imho. Commented Jun 5, 2013 at 19:32

The function you need is TrigToExp, e.g.

TrigToExp @ ArcTan[x]

 1/2 I Log[1 - I x] - 1/2 I Log[1 + I x]


There is an inverse function for TrigToExp, namely ExpToTrig

ExpToTrig[ 1/2 I Log[1 - I x] - 1/2 I Log[1 + I x]]

ArcTan[x]


They both are Listable:

Attributes @ {ExpToTrig, TrigToExp}

{{Listable, Protected}, {Listable, Protected}}


Therefore you can use it e.g. this way:

TrigToExp [Sin[x] < x < Tan[x] ]

 1/2 I E^(-I x) - 1/2 I E^(I x) < x < (I (E^(-I x) - E^(I x)))/(E^(-I x) + E^(I x))