Infinite Series - TCD Maths home
... log(1) = 0, | Im log(z)| < π and exp(log(z)) = z for all z ∈ D0 . This function has the property that log(reiθ ) = loge r + iθ for all real numbers r and θ satisfying r > 0 and −π < θ < π, where loge r denotes the natural logarithm of the real number r. Proof Given any complex number w belonging to ...
... log(1) = 0, | Im log(z)| < π and exp(log(z)) = z for all z ∈ D0 . This function has the property that log(reiθ ) = loge r + iθ for all real numbers r and θ satisfying r > 0 and −π < θ < π, where loge r denotes the natural logarithm of the real number r. Proof Given any complex number w belonging to ...
Review of Basic Concepts
... In this chapter, we review a variety of basic mathematical concepts that will be needed to discuss calculus. All of these topics are treated very super cially, as our goal is only to provide a brisk review of the necessary material. ...
... In this chapter, we review a variety of basic mathematical concepts that will be needed to discuss calculus. All of these topics are treated very super cially, as our goal is only to provide a brisk review of the necessary material. ...
Chapter Two
... If a function f (x) is not continuous at a point c, we say that f (x) is discontinuous at c and call c a point of discontinuity of f (x). The Continuity Test The function y=f(x) is continuous at x=c if and only if the following statements are true:1. f (c) exists (c lies in the domain of f). 2. lim ...
... If a function f (x) is not continuous at a point c, we say that f (x) is discontinuous at c and call c a point of discontinuity of f (x). The Continuity Test The function y=f(x) is continuous at x=c if and only if the following statements are true:1. f (c) exists (c lies in the domain of f). 2. lim ...
-7- 6CCM320A (CM320X) SECTION D – Distribution of Prime
... i. Prove that there exists precisely one prime number p for which the 4 numbers p + 6, p + 12, p + 18 and p + 24 are also primes, and find it. ii. Prove that there exist infinitely many primes p such that p + 6 is composite. You may use any result presented in class. (Hint: You may either want to ar ...
... i. Prove that there exists precisely one prime number p for which the 4 numbers p + 6, p + 12, p + 18 and p + 24 are also primes, and find it. ii. Prove that there exist infinitely many primes p such that p + 6 is composite. You may use any result presented in class. (Hint: You may either want to ar ...
The Euler characteristic of the moduli space of curves
... Let F be a closed, oriented surface of genus g with basepoint p. The set of isotopy classes of orientation preserving homeomorphisms of F which fix p is a group under composition called the mapping class group and is denoted F~1. The Teichmfiller space ~1 is the space of all conformal equivalence cl ...
... Let F be a closed, oriented surface of genus g with basepoint p. The set of isotopy classes of orientation preserving homeomorphisms of F which fix p is a group under composition called the mapping class group and is denoted F~1. The Teichmfiller space ~1 is the space of all conformal equivalence cl ...
Math 416 – Introduction to Abstract Algebra
... Number of generators Number of subgroups of each size Number of elements of each order Groups are the same are called isomorphic. Rigorously, Two groups, G, * and G’, ∙ are isomorphic if there exists a 1-1 and onto function, φ: G G’ such that, for any elements a, b G, φ(a*b) = φ(a)∙ φ(b) Example ...
... Number of generators Number of subgroups of each size Number of elements of each order Groups are the same are called isomorphic. Rigorously, Two groups, G, * and G’, ∙ are isomorphic if there exists a 1-1 and onto function, φ: G G’ such that, for any elements a, b G, φ(a*b) = φ(a)∙ φ(b) Example ...
Foundations of the golden ratio base
... This follows directly from the division algorithm, with divisor 2, extended to allow negatives. So, for −5 · φk we would have −1(2 · 2 + 1)φk . 2. Add m to the digits k + 1 and k − 2. This follows from (8). Change the k digit to sn. This is the remainder of removing all the even factors from c. Repe ...
... This follows directly from the division algorithm, with divisor 2, extended to allow negatives. So, for −5 · φk we would have −1(2 · 2 + 1)φk . 2. Add m to the digits k + 1 and k − 2. This follows from (8). Change the k digit to sn. This is the remainder of removing all the even factors from c. Repe ...
Full text
... that x + y is a prime greater than p . If x 4- y is composite, it must have a prime divisor greater than p. . This last statement follows from the fact that every prime q<.Vk divides m and hence divides x. If q divides x + y, then it divides y9 which contradicts the fact that (x9 y, 2) is a primitiv ...
... that x + y is a prime greater than p . If x 4- y is composite, it must have a prime divisor greater than p. . This last statement follows from the fact that every prime q<.Vk divides m and hence divides x. If q divides x + y, then it divides y9 which contradicts the fact that (x9 y, 2) is a primitiv ...
05 FX115 Ex Cmplx Num
... Have students examine a table of values for an exponential function. Using the table function of the fx-115ES, have them input the function f(x) = 2x. Set a start value of –5 and an ending value of 5 with a step increment of 1. Ask: What does the shape of an exponential function look like? When you ...
... Have students examine a table of values for an exponential function. Using the table function of the fx-115ES, have them input the function f(x) = 2x. Set a start value of –5 and an ending value of 5 with a step increment of 1. Ask: What does the shape of an exponential function look like? When you ...
