How to tell where parentheses go in functional notation? The professor gave us a function $f(z) = \ln r + i \theta$ (this is for a complex analysis class). He doesn't like answering students' questions and there's no assigned textbook so I don't know where to look up such a function. How can I tell where the parentheses are supposed to go for this equation?
 A: Dropping the parentheses when using functional notation is done often with the trigonometric functions and logarithms, à la $\sin x$ or $\log x$. This can (and does) lead to confusion, as we can see.
Usually terms which are multiplied together are all part of the function argument, while terms added or subtracted are not:
$$\log a + b =\log(a)+b\\
\log a - b =\log(a)-b\\
\log ab + c = \log\left(ab\right)+c\\
\log\frac ab + c = \log\left(\frac ab\right)+c\\
$$
On the other hand:
$$a\log b = a\log(b)\\
\frac{\log a}b = \frac{\log(a)}b$$
By these conventions, your equation means $f(z) = \ln(r) + i \theta$.

Note that there are some exceptions. For example, $\sin x\cos x$ probably means $\sin(x)\cos(x)$ rather than $\sin(x\cos(x))$, while $\sin\omega t$ probably means $\sin(\omega t)$ rather than $\sin(\omega)t$. For reasons such as this, it is good to always use enough parentheses to make the expression unequivocal.
A: In addition to syntactic answers already discussed, you can think a little about which makes the most sense in context.  For a complex analysis class, $f(z) = \ln(r) + i\theta$ is a common function to encounter.  If $z$ is a complex number with modulus $r$ and principal argument $\theta$, then $f(z)$ is the principal branch of the complex logarithm, a function which is used all the time.  On the other hand, $f(z) = \ln(r + i\theta)$ doesn't seem to be a particularly useful function.
A: With an expression like this, we have three possibilities:
Either
\begin{align*}
f(x) &= \ln(r) + i\theta,\\
f(x) &= \ln(r + i)\theta, \text{ or}\\
f(x) &= \ln(r + i\theta).
\end{align*}
We can assume the professor means $\ln(r) + i\theta$, since it requires the simplest expression in parenthesis. Generally, if the expression in parenthesis is at all 'complicated' (requires any binary operations like addition, subtraction, etc), it belongs in parenthesis. If no such operations are required, parenthesis can be safely omitted and we agree to default to the simplest argument possible for the function.
That's not to say I agree with saying something is 'obvious', but I'm merely providing hints for the future!
