Capital or lowercase letter after a formula?

Question

Often when creating reports for labs or something like that I find myself writing something like:

Question is:

As the formula is often on its own line, once I say "(...) where e (...)" should the 'W' be capital or lowercase? What I mean is that changing from formula back to text is like starting a new line.

Additional question (if I am allowed):

In the preamble to the formula "(...) for any real number x (...)", should I put a colon (:), a comma (,), a dot (.), or nothing (as it is in this question)?

The possibilities are

• "(...) for any real number x, e^ix (...)", f(x)=(z-y)e^0.01-0.01x+y

• "(...) for any real number x: e^ix (...)",

• "(...) for any real number x. e^ix (...)"

• and as it is in the question (no punctuation after "x").

Note: In wikipedia they use a dot.

Answer 1

Personally (and in accordance with the conventions that I've seen in published works), I'd write it as follows:

Euler's formula states that, for every real number~$x$,

\begin{equation}
  e^{ix} = \cos x + i \sin x,
\end{equation}

where $e$~is the base of the natural logarithm.

The philosophy is that the equation is a statement, part of the sentence. You should therefore punctuate it in the same way as you would were it run in with the text.

See also this MathOverflow thread on the subject. In particular, this answer has a few examples in this style.

Here's another good answer on TeX.SE.

Answer 2

for any real number $x$

$e^{ix}=\cos{x}+i\sin{x}$

where $e$ is the base of the natural logarithm (...)"

I can see no reason for capitalising 'w' in that situation - a newline doesn't signal the start of a new sentence. I would find it very confusing if you capitalised it.

Answer 3

Only if it's two different sentences. If the formula is just the subject or the object of the same sentence then there should be no need to capitalize the next word. (unless it is a proper noun or a quote or something like that).