The statement about pointwise convergence follows because C is a complete metric space. If fn → f uniformly on S, then |fn(z) − fm(z)| ≤ |fn(z) − f(z)| + |f(z) − fm(z)|, hence {fn} is uniformly Cauchy. Conversely, if {fn} is uniformly Cauchy, it is pointwise Cauchy and therefore converges pointwise to a limit function f. If |fn(z)−fm(z)| ≤ ε for all n,m ≥ N and all z ∈ S, let m → ∞ to show that |fn(z)−f(z)|≤εforn≥N andallz∈S. Thusfn →f uniformlyonS.
2. This is immediate from (2.2.7).
3. We have f′(x) = (2/x3)e−1/x2 for x ̸= 0, and f′(0) = limh→0(1/h)e−1/h2 = 0. Since f(n)(x) is of the form pn(1/x)e−1/x2 for x ̸= 0, where pn is a polynomial, an induction argument shows that f(n)(0) = 0 for all n. If g is analytic on D(0,r) and g = f on (−r,r), then by (2.2.16), g(z) =
Answer:
<h2>
E = 2.8028*10⁻¹⁹ Joules</h2>
Explanation:
The minimum energy needed to eject electrons from a metal with a threshold frequency fo is expressed as E = hfo
h = planck's constant
fo = threshold frequency
Given the threshold frequency fo = 4.23×10¹⁴ s⁻¹
h = 6.626× 10⁻³⁴ m² kg / s
Substituting this value into the formula to get the energy E
E = 4.23×10¹⁴ * 6.626 × 10⁻³⁴
E = 28.028*10¹⁴⁻³⁴
E = 28.028*10⁻²⁰
E = 2.8028*10⁻¹⁹ Joules
Cosmic, or background, radiation is the small amount of high energy radiation which is mostly left over from the big bang or from supernovas. It is mostly single protons, but also alpha particles and even sometimes heavier elements. It can also refer to the low levels of electromagnetic radiation present all over the universe.
