# Thread: Principle value of argument

1. ## Principle value of argument

How do I find the principle value of the argument of ?

2. Originally Posted by RossLeeds
How do I find the principle value of the argument of ?
Principal (note spelling) value of the argument of what?

3. Originally Posted by Opalg
Principal (note spelling) value of the argument of what?
Sorry, confusion with the image - the princiPAL value of the argument of:

exp( 7 + 4i )

4. Originally Posted by RossLeeds
Sorry, confusion with the image - the princiPAL value of the argument of:

exp( 7 + 4i )
$e^{7 + 4i} = e^7 e^{4i}$

So the argument is 4 radians. Therefore the principal argument is ....?

5. Originally Posted by mr fantastic
$e^{7 + 4i} = e^7 e^{4i}$

So the argument is 4 radians. Therefore the principal argument is ....?
arctan 4?

6. Originally Posted by RossLeeds
arctan 4?
No. The argument is an angle (in this case, 4 radians). But an angle is only determined up to a multiple of 2π. The principal value of the argument is obtained by taking the value of the argument that lies in some specified interval of length 2π, usually the interval $(-\pi,\pi]$. So if 4 lies in that interval then the principal value is just 4. Otherwise, you'll have to add or subtract a multiple of 2π to 4 in order to bring it into that interval.