# All Educational Materials for Mathematics at Harvard University

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## MATH 113 Harvard 113 Fall 01113FinalSolution

6 Page
15 Feb 2019
Final examination solutions: (15 points, state the cauchy-riemann equations for the real and imaginary parts of an analytic function. Suppose f = u + i
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15 Feb 2019
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## MATH 113 Harvard 113 Fall 01113review

3 Page
15 Feb 2019
The nal exam will cover everything in chapters 1-11 and 13 of the course text, excluding a few results we skipped, e. g. the schwarz re ection prin- ci
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HARVARDAllFall

## MATH 113 Harvard 113 Fall 01113Solution3

2 Page
15 Feb 2019
Since eiy = cos y + i sin y, one sees that for any real y0, eiy maps the interval [y0, y0 + 2 ) bijectively onto the unit circle. Now ex maps the real
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HARVARDAllFall

## MATH 113 Harvard 113 Fall 01113Solution2

3 Page
15 Feb 2019
October 11, 2001: find all complex solutions of ez = 5: ex+iy = ex(cosy + isiny) = 5(cos + isin ). Note: if er1cis 1 = er2cis 2 then r1 = r2 and 1 2 =
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HARVARDAllFall

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15 Feb 2019
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HARVARDAllFall

## MATH 1B Harvard Solutions2s99

18 Page
15 Feb 2019
Rules: write all answers in the spaces provided below. Make sure that in each problem you have at least one sentence explaining the key idea. Problem 1
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HARVARDAllFall

## MATH 1B Harvard sol Exam1 f04

9 Page
15 Feb 2019
Rules: write all answers in the spaces provided below. Make sure that in each problem you have at least one sentence explaining the key idea. Problem 1
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HARVARDAllFall

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15 Feb 2019
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HARVARDAllFall

## MATH19 Harvard Midterm1F04sol19 Fall 04

14 Page
15 Feb 2019
\bu have two hours to take this midtcrm. Anv ansrver in dccirnal forrn nrust bc accuratc to threc decirnal places. urtless otherwise specificd. Pace l.
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## Mathematics MATH E-311 Lecture 17: application of laplace transformation

8 Page
27 Apr 2019
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## Mathematics MATH E-311 Lecture 90: power series representation of functions

3 Page
27 Apr 2019
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## Mathematics MATH E-311 Lecture 13: power of complex numbers

7 Page
27 Apr 2019
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## Mathematics MATH E-311 Lecture 15: power and roots of complex numbers

7 Page
27 Apr 2019
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## Mathematics MATH E-311 Lecture 13: periodic expansion

6 Page
27 Apr 2019
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## Mathematics MATH E-311 Lecture 17: complex numbers

6 Page
27 Apr 2019
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## Mathematics MATH E-311 Lecture Notes - Lecture 17: Horse Length

6 Page
27 Apr 2019
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## Mathematics MATH E-311 Lecture 90: power series representation of functions

3 Page
27 Apr 2019
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## Mathematics MATH E-311 Lecture Notes - Lecture 17: Inverse Trigonometric Functions, Radian

4 Page
27 Apr 2019
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27 Apr 2019
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6 May 2016
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1 Apr 2016
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## Computer Science CSCI E-51 Lecture Notes - Lecture 4: Abstract Syntax Tree

3 Page
21 Feb 2015
Simple example let inc = (fun x -> x+1) in inc 41 (fun x -> x+ 1) 41. Next, we are going to look at code that does this evaluation for us. A value is s
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1 Apr 2016
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4 Feb 2014
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8 Aug 2016
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21 Feb 2017
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14 Mar 2014
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10 Feb 2014
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1 Apr 2016
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10 Feb 2014
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## Chapter 18 - The Markets for the Factors of Production

1 Page
10 Feb 2014
Marginal product of labor: increase in the amount of output from an additional unit of labor diminishing marginal product: marginal product of input de
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2 Page
10 Feb 2014
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## Economics 10a Chapter Notes - Chapter 10: Externality

2 Page
10 Feb 2014
Maximize welfare of society (cid:224) point where demand curve crosses social cost curve equilibrium quantity larger than socially optimal quantity for
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2 Page
21 May 2014
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## OEB 53 Chapter Notes -Selective Breeding

1 Page
17 Feb 2014
The origin of the species: chapter 5 (laws of variation) The effects of use and disuse seem to puzzle darwin. Darwin is unsure whether change is a resu
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21 May 2014
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17 Feb 2014
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29 Nov 2016
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HARVARDWinter

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6 May 2016
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29 Nov 2016
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29 Nov 2016
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1 Apr 2016
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## [12a] - Final Exam Guide - Everything you need to know! (580 pages long)

580 Page
29 Nov 2016
The position vector r of the point p. The components x, y, z of the position vector r. The components ax, ay, az of an arbitrary vector a. Vector a in
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10 Feb 2014
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29 Nov 2016
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## Computer Science CSCI E-51 Lecture Notes - Lecture 4: Abstract Syntax Tree

3 Page
21 Feb 2015
Simple example let inc = (fun x -> x+1) in inc 41 (fun x -> x+ 1) 41. Next, we are going to look at code that does this evaluation for us. A value is s
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74 Page
29 Nov 2016
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261 Page
30 Nov 2019
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2 Nov 2019
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2 Nov 2019
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2 Nov 2019
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10 Jun 2019
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10 Jun 2019
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10 Jun 2019
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10 Jun 2019
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10 Jun 2019
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## Physical Sciences 3 Lecture Notes - Lecture 11: Cross Product, Scandinavian Mountains, Bnet

12 Page
10 Jun 2019
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