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### prob0

Course: MATH 503, Fall 2009
School: Rutgers
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Word Count: 335

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503 Problem Math Set 0 9/4/2007 Solutions are due at the beginning of class on Friday, September 7, 2007. 1. In this problem, A is a non-empty subset of R. A real-valued function g defined on A is bounded if g = sup |g(x)| &lt; . Also in this problem, {fn }n1 is a sequence of xA bounded real-valued functions defined on A with bn = fn . a) Suppose that {fn } converges uniformly on A to a function, f ....

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503 Problem Math Set 0 9/4/2007 Solutions are due at the beginning of class on Friday, September 7, 2007. 1. In this problem, A is a non-empty subset of R. A real-valued function g defined on A is bounded if g = sup |g(x)| < . Also in this problem, {fn }n1 is a sequence of xA bounded real-valued functions defined on A with bn = fn . a) Suppose that {fn } converges uniformly on A to a function, f . Prove that f is bounded, and, if b = f , b = lim bn . n b) Suppose that {fn } converges pointwise on A to a function, f . Must f be bounded? Either prove that f is bounded, or give a counterexample. c) Suppose that {fn } converges pointwise on A to a bounded function, f , and that b = f . Must b = lim bn ? Either prove that this equality is correct, or give a n counterexample. 2. Part of an open cone with vertex the origin and aperture or opening is shown in the diagram to the Give right. a precise definition (in either R2 or C) of such an object (it is open, so only the inside of the cone should be included!). Fix in (0, ), and suppose W is an open cone with aperture . Prove the following reverse triangle inequality (using algebraic manipulation in either R2 or C): There is C > 0 depending on (and not on W ) so that if z1 and z2 are in W , then C (|z1 | + |z2 |) |z1 + z2 |. 3. Suppose {tn }n1 is a sequence of positive real numbers. Prove that there is a power series n=0 x is greater than tn whe...

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Rutgers - MATH - 503
Math 503 Name E-mail addressStudent informationPlease print!I would like to make a course webpage with the information above displayed. This should increase the ability of students to work together in this course. Do you agree to having this in
Rutgers - MATH - 135
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Rutgers - MATH - 135
Rutgers - MATH - 135
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Rutgers - MATH - 135
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Rutgers - MATH - 135
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Rutgers - MATH - 135
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Rutgers - MATH - 135
Rutgers - MATH - 135
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Rutgers - MATH - 135
Rutgers - MATH - 135
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Rutgers - MATH - 135
%!PS-Adobe-2.0 %Creator: dvips 5.528 Copyright 1986, 1994 Radical Eye Software %Title: syl.dvi %CreationDate: Sun Aug 31 20:49:34 1997 %Pages: 1 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %EndComments %DVIPSCommandLine: dvips syl.dvi -o syl.ps %DVI
Rutgers - MATH - 135
%!PS-Adobe-2.0 %Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %Title: review1.dvi %CreationDate: Fri Sep 26 08:52:34 1997 %Pages: 2 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %EndComments %DVIPSCommandLine: dvips -i review1.dvi -o r
Rutgers - MATH - 135
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Rutgers - MATH - 135
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Rutgers - MATH - 135
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Rutgers - MATH - 135
Rutgers - MATH - 135
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Rutgers - MATH - 135
Rutgers - MATH - 135
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Rutgers - MATH - 403
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Rutgers - MATH - 403
Math 403:01Student informationPlease print! Name Major(s) E-mail address I would like to make a course webpage with the information above displayed. This should increase the ability of students to work together in this course. Do you agree to havi
Rutgers - MATH - 403
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Rutgers - MATH - 403
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Rutgers - MATH - 403
%!PS-Adobe-2.0 %Creator: dvips(k) 5.83 Copyright 1998 Radical Eye Software %Title: sketch_zs.dvi %Pages: 1 %PageOrder: Ascend %BoundingBox: 0 0 596 842 %DocumentFonts: CMR10 CMBX10 CMTI10 CMMI10 CMR7 CMSY10 %EndComments %DVIPSWebPage: (www.radicaleye
Rutgers - MATH - 403
January 28, 2001 Math 403, section 1Locate some complex numbersAt least 2 students and no more than 4 students must work on this problem. Please print the names of those working on this problem:z1Suppose z is the complex number pictured. Dra
Rutgers - MATH - 403
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Rutgers - MATH - 403
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Rutgers - MATH - 403
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Rutgers - MATH - 403
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Rutgers - MATH - 403
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Rutgers - MATH - 403
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Rutgers - MATH - 403
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Rutgers - MATH - 403
Math 403, section 1Questions about exp and logFebruary 11, 2002Exactly 2 students should work on this. Their names should be printed below.expWrite all values of exp(1 + i) :105Show all complex zs having a value of exp z which is real o
Rutgers - MATH - 403
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Rutgers - MATH - 403
Math 403, section 1Answers about exp and log expFebruary 12, 2002Since exp z = eRe z (cos(Im z) + i sin(Im z), we see that exp(1+i) must be e1 (cos 1)+i(sin 1). Numerical approximations of the numbers are not expected. The complex zs which have
Rutgers - MATH - 403
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Rutgers - MATH - 403
Math 403:01 course expectationsHeres what I expect from you in this course, beginning with an observation about myself. The statements below may help you make a reasonable judgment about the time and eort needed for success in this course. 0. I will
Rutgers - MATH - 403
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Rutgers - MATH - 403
Rutgers - MATH - 403
Rutgers - MATH - 403
Rutgers - MATH - 403
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Rutgers - MATH - 403
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Rutgers - MATH - 403
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Rutgers - MATH - 403
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Rutgers - MATH - 403
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Rutgers - MATH - 403
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Rutgers - MATH - 403
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Rutgers - MATH - 403
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Rutgers - MATH - 403
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Rutgers - MATH - 403
Rutgers - MATH - 403
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Rutgers - MATH - 403
Rutgers - MATH - 403
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Rutgers - MATH - 403
Rutgers - MATH - 403
%!PS-Adobe-2.0 %Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %Title: exam2.dvi %Pages: 8 %PageOrder: Ascend %BoundingBox: 0 0 612 792 %EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips exam2.dvi -o exam2.ps %DVIPS
Rutgers - MATH - 403
Rutgers - MATH - 403
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Rutgers - MATH - 403
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Rutgers - MATH - 403
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