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I have a question about the conserved charge for the ghost number current in $bc$ conformal field theory in Polchinski's string theory p62. It is said

> 
> For the ghost number current (2.5.14), $j=-:bc:$, the charge is 
>  $$ N^g= \frac{1}{2 \pi i} \int\_0^{2 \pi} dw j\_w = \sum\_{n=1}^{\infty} ( c\_{-n} b\_n -b\_{-n} c\_n ) + c\_0 b\_0 - \frac{1}{2} \,\,\,\, (2.7.22) $$
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First, what is $j\_w$? Seems it haven't been defined before. Is current (2.5.14) in variable w, i.e. $z=\exp(-iw)$? Then how to derive Eq. (2.7.22)? I tried to derive it but differs the constant term and lots of (conformal) normal ordering
$$ N^g= \frac{1}{2 \pi i} \int\_0^{2 \pi} dw j\_w = \frac{1}{2 \pi i} \oint dz j\_z $$
$$ = -\frac{1}{2 \pi i} \oint dz :bc: =\frac{-1}{2 \pi i} \oint dz \sum\_{m=-\infty}^{\infty} \sum\_{n=-\infty}^{\infty} \frac{ : b\_m c\_n:}{z^{m+n+1}} (\mathrm{omit} \,\,\, \lambda \,\,\, \mathrm{in} \,\,\, \mathrm{2.7.16})$$
$$ = - \sum\_{m=-\infty}^{\infty} \sum\_{n=-\infty}^{\infty} : b\_m c\_n: \delta\_{m+n,0} = - \sum\_{m=-\infty}^{\infty} : b\_m c\_{-m}: $$
$$ =- :b\_0 c\_0: - \sum\_{m=-\infty}^{-1} : b\_m c\_{-m}: - \sum\_{m=1}^{\infty} : b\_m c\_{-m}: = -:b\_0 c\_0: - \sum\_{m=1}^{\infty} : b\_{-m} c\_{m}: - \sum\_{m=1}^{\infty} : b\_m c\_{-m}: $$
$$ = :c\_0b\_0: - 1 - \sum\_{m=1}^{\infty}: b\_{-m} c\_{m}: - : c\_{-m} b\_{m}: - \sum\_{m=1}^{\infty} \{ :b\_{m},c\_{-m}: \} $$
$$ = \sum\_{n=1}^{\infty} (:c\_{-n} b\_n: - :b\_n c\_{-n}:) + :c\_0b\_0: - \infty$$ 
$$ = \sum\_{n=1}^{\infty} (\circ :c\_{-n} b\_n: \circ- \circ :b\_n c\_{-n}: \circ) + \circ :c\_0b\_0: \circ - \infty$$ 
$\circ \cdots \circ$ indicates creation-annihilation normal ordering

I guess there is huge problem in my derivation, but how to get (2.7.22) correctly?