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.