Homework Statement
Hello I am trying to solve the following Differential Equation:
r^2\frac{d^2R}{dr^2}+r\frac{dR}{dr}-\left[A^2r^4-B^2r^2-C^2]R=0
where A,B and C are constants-
Homework Equations
I have read this equation is calle "Bessel wave eq" but I can't find the reference...
Homework Statement
Show that if f is continuously differentiable on [a, b], then there is a sequence of
polynomials pn converging uniformly to f such that p'n converge uniformly to f' as
well.
Homework Equations
The Attempt at a Solution
Let pn(t) = cn t^n
Use uniform convergence and integrate...
Homework Statement
Show that if f is continuously differentiable on [a, b], then there is a sequence of
polynomials pn converging uniformly to f such that p'n converge uniformly to f' as
well.Homework Equations
The Attempt at a Solution
Let pn(t) = cn t^n
Use uniform convergence and integrate...
This one doesn't have an antiderivative so you have to express first (exp(x)-1)/x as a Taylor series and then integrate term by term, and you have to make sure that the resultant series converges.
You already know the series expansion of exp(x), so it's easy to find the series of exp(x)/x just...
Homework Statement
f nonnegative continuous function on the interval [0,1]. Let M be the supremum of f on the interval. Prove:
\lim_{n \rightarrow \infty} \left[ \int^1_0 f(t)^n dt \right] ^{1/n} = M
Homework Equations
The Attempt at a Solution
I was trying using Upper Sums but...
Homework Statement
For f nonnegative and continuous on [0,1], prove.
\left( \int f \right) ^2 < \int f^2
With the limits from 0 to 1.
Homework Equations
The Attempt at a Solution
I was trying to use Upper sums, i.e.
\inf \sum \Delta x_i M_i(f^2) = \inf \sum \Delta x_i...