Archives for Pure Math - Page 2
Suppose cp is a C1 function on R such that cp(x)+a and cp’(x)~b as xjoo. Prove or give a counterexample: b must be zero.
Suppose cp is a C1 function on R such that cp(x)+a and cp’(x)~b as xjoo. Prove or give a counterexample: b must be zero. To address the question, we need…
what is the langlands program?
The Langlands program is a set of far-reaching and deep conjectures proposed by Robert Langlands in 1967, which aims to relate and unify various areas of mathematics, including number theory,…
Let R > 1 and let f be analytic on IzI < R except at z = 1, where f has a simple pole. If M n=O is the Maclaurin series for f, show that hn,+,M alL exists.
To show that the limit limn→∞nan\lim_{n \to \infty} n a_nlimn→∞nan exists for the Maclaurin series f(z)=∑n=0∞anznf(z) = \sum_{n=0}^{\infty} a_n z^nf(z)=∑n=0∞anzn of the function fff, which is analytic in ∣z∣<R|z| <…
An accurate map of California is spread out flat on a table in Evans Hall, in Berkeley. Prove that there is exactly one point on the map lying directly over the point it represents.
To prove that there is exactly one point on the map of California that lies directly over the point it represents, we can use a combination of the Brouwer Fixed-Point…
Let f : R + R be continuous, with 00 s_, If(x)l dx < o. Show that there is a sequence (x,,) such that x, -+ 00, x, f (x,) 4 0, and x, f(-xc,) 4 0 as n -+ o.
Let f : R + R be continuous, with 00 s_, If(x)l dx < o. Show that there is a sequence (x,,) such that x, -+ 00, x, f (x,)…
Given conditions: lim 𝑡 → 𝑎 𝑔 ( 𝑡 ) = 𝑏 lim t→a g(t)=b lim 𝑥 → 𝑏 𝑓 ( 𝑥 ) = 𝑐 lim x→b f(x)=c To prove: lim 𝑡 → 𝑎 𝑓 ( 𝑔 ( 𝑡 ) ) = 𝑐 lim t→a f(g(t))=c
Given conditions: limt→ag(t)=b\lim_{t \to a} g(t) = blimt→ag(t)=b limx→bf(x)=c\lim_{x \to b} f(x) = climx→bf(x)=c To prove: limt→af(g(t))=c\lim_{t \to a} f(g(t)) = climt→af(g(t))=c Using the definition of the limit: limt→ag(t)=b\lim_{t \to…