How can I get Mathematica to write Log instead of ArcTan?

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
– Artes
Jun 5 ’13 at 16:05



That’s awesome, thanks a lot man. You should post this as an answer.
– mapel
Jun 5 ’13 at 16:06



So now I’m curious. What’s bothering you in ArcTans?
– Sjoerd C. de Vries
Jun 5 ’13 at 18:03



@Vries They are way too restricted for me, log\logs are much more general and easier to work with imho.
– mapel
Jun 5 ’13 at 19:32


1 Answer


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]]


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))