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.

 

[blue_raver22]

The value of the norm |x|_p in p-adic analysis depends on the specific context and values given in the problem. In general, the p-adic norm can be defined as the inverse of the p-adic valuation, which measures the highest power of p that divides a given number. Therefore, the value of the norm |x|_p will change depending on the values of x and p.

a) If x=0 and p is different from 0, then the norm |x|_p will be equal to 0. This is because the p-adic norm of 0 is defined as the inverse of the p-adic valuation, which in this case would be 0.

b) If x=0 and p=0, then the norm |x|_p is undefined. This is because the p-adic norm is only defined for nonzero values of p.

c) If x=0 and p=∞, then the norm |x|_p is also undefined. This is because in p-adic analysis, the p-adic norm is only defined for prime values of p.

d) If x is a real number, then the p-adic norm will be equivalent to the standard absolute value of x. This is because the p-adic norm can be extended to the real numbers in a way that is consistent with the standard absolute value.

e) If x is a rational number and p is infinite, then the norm |x|_p will be equal to 1. This is because in p-adic analysis, the infinite prime p is treated as a limit of the finite primes, and the norm |x|_p converges to 1 as p approaches infinity.

In terms of evaluating the integral over Q_p of ∫_Q_p |x|_p f(x), the value of the norm |x|_p will affect the convergence of the integral. If the norm |x|_p is equal to 1, the integral may converge, but if it is equal to 0, the integral will diverge. The specific value of the integral will depend on the function f(x) and the values of x and p.