advanced calculus
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fbel
Mathematics
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I want the answer for three question about integration ..
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Since differentiation is also a limit operation, this is another “interchange of
limit operations” result.
D2f is uniformly continuous on A x J.
Exercises
3.1 (a) If f is integrable, prove that f? is integrable. Hint: Given € > 0, let h and k be
step functions such that hf $k and S (k - h) < /M, where M is the maximum value
of \k(x) + h(x). Then prove that h2 and kare step functions with h? Sf? Sk? (we
may assume that I shfk since f is integrable if and only if \f| is--why?), and
that ſ (k2 – ha)< €. Then apply Theorem 3.3.
(b) If f and g are integrable, prove that fg is integrable. Hint: Express fg in terms of
(f+g)2 and (f-g)2; then apply (a).
3.2 Let f: R → R be a bounded function which vanishes outside the interval Q. Show that
fis integrable with Sf=I if and only if
lim R(f, PK, Ix)=1
k-00
for every sequence of partitions {Pulis and associated selections {Svi such that
limk>. (mesh of Pk)=0.
3.3 Let A and B be contented sets, and f: Ax BA a uniformly continuous function.
If : A R is defined by
11 YEA
Textbook, Page 233-234, 3.1, 3.7, plus,
1. Consider two intervals A = (0,2) > [1,4 and B = [1,3] x [0,3) in R. Let
f = a4A and g = b4B, where 4A and B are the characteristic functions of A and
B, respectively, and a, b are constants.
(a) Find a step function expression for f +g.
(a) Evaluate the integralſ (8 + 9).
3.7. For each positive integer n, the Bessel function
Jn't may be defined by
In W)
cosat at
S 1. tj 1/2
103.5
. (2n-1)
1) T -
prove that Jn (x) Satisfies Bessel's differential
equation
-)Jn
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. 14
1
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