What is the Value of the Norm |x|_p in P-adic Analysis?

[zetafunction]

in p-adic analisis what is the value of the norm $$|x|_{p}$$

a) x=0 and p is different from 0

b) x=0 and P=0

c) x=0 and $$p=\infty$$

d) x is a real number

e) x is a Rational number and p is infinite

how i evaluate the integral over $$Q_{p}$$ of $$\int_{Q_{p}} |x|_{p}f(x)$$

 

[g_edgar]

When you say $p$-adic analysis, $p$ is a prime, so $p=0$ is not used. $|0|_p = 0$. Sometimes the usual absolute value $|x|$ is called the $\infty$-adic absolute value, and $\infty$ is listed among the "primes". The $p$-adic absolute value is defined for the $p$-adic numbers, not the real numbers. Except the $\infty$-adic numbers may mean the real numbers. For your integral, I suppose we use the Haar measure.

 

[zetafunction]

yes i use Haar measure type i think it was $$\frac{p}{p-1}|x|_{p}$$ so for p=infinite it becomes 1/x

should i expand f into a power series and then integrate term by term to get the p-adic integral?

 

[g_edgar]

Power series is probably not useful. Your integrand $$|x|_p$$ has only countably many values, and integrals of that kind are best converted to sums.