Notes for Analysis Et/Wi GS TU Delft 2001 5/11/2019
Week 1. Complex numbers a. 5/11/2019
Week 1. Complex numbers b. 5/11/2019
Week 1. Complex numbers c. 5/11/2019
Week 1. Complex numbers d. Only for n4 the roots can be computed algebraically. For n=2 by the abc-formula. For n=3 by Cardano’s method. For n=4 by Ferrari’s method. 5/11/2019
Week 2. Limit, the definition 5/11/2019
Week 2. Derivative, the definition 5/11/2019
Week 2. Application of Implicit Differentiation 5/11/2019
Week 2. Special functions 5/11/2019
Week 3. Mean Value Theorem a. 5/11/2019
Week 3. Mean Value Theorem b. 5/11/2019
Week 3. Antiderivatives and Integrals Antiderivative ‘inverse derivative’ Integral ‘signed area under graph’ 5/11/2019
Week 3. The Integral define integral for step-functions by the ‘signed’ surface area of the rectangles approximate function f by sequence of stepfunc-tions { sn(x) } define integral for f by the limit of the integrals for sn(x) Only for rectangles we have an elementary formula for the surface area: length width. 5/11/2019
Week 3. Fundamental Theorem of Calculus Although defined in completely different ways there is a well-known relation between antiderivative and integral. The relation is stated in this theorem: I: from integral to anti-derivative II: from antiderivative to integral 5/11/2019
Week 3. Substitution Rule 5/11/2019
Week 3. Partial Integration Rule 5/11/2019
Week 3. Both rules in shorthand If one doesn’t know why, just magic remains. 5/11/2019
Week 3. Partial Integration, an example Not all v’s are equal. 5/11/2019
Week 4. Integration of Rational Functions, a. 5/11/2019
Week 4. Integration of Rational Functions, b. 5/11/2019
Week 4. Division in Rational Functions Hence we only have to consider rational functions with the denominator of lower degree than the enumerator. 5/11/2019
Week 4. Splitting of Rational Functions 5/11/2019 But what about complex roots?
Week 4. Splitting of Rational Functions, complex roots To understand the above one has to take a course on Complex Functions. For the moment we combine complex factors and proceed (in)directly. It appears that we can always combine complex (non-real) fractions pairwise to real fractions. 5/11/2019
Week 4. Integration of Rational Functions, a recipe Although the factorisation always exists, it is not always algebraically computable. Usually the last two steps are more convenient the other way around. One may also proceed without complex numbers. 5/11/2019
Week 4. Integration of Rational Functions, an example Complex Real 5/11/2019
Week 4. Integration of Rational Functions, another example After dividing out, finding roots, splitting fractions, computing constants, combining, substitution, one finally may integrate. 5/11/2019
Week 4. Improper Integrals, a. Integrals are defined through approximations by stepfunctions with increasing but finitely many steps. There is no such direct approximation in the following cases: Unbounded interval Unbounded function Solution: define the improper integral by a limit. 5/11/2019
Week 4. Improper Integrals, b. Unbounded interval 5/11/2019
Week 4. Improper Integrals, c. Unbounded function : 5/11/2019
Week 4. Improper Integrals, Comparison Test A similar comparison test can be formulated for improper integrals of the second type. 5/11/2019
Week 4. Special Improper Integrals What about ? The improper integrals above are often candidates for the comparison test. 5/11/2019
Week 5. Differential Equations A differential equation gives a relation between a function and its derivatives, for example: Aim: derive properties of the solution or, if possible, give even a closed formula for the solution. often initial values are given such as: and usually an nth -order d.e. needs n initial conditions to have precisely one solution. 5/11/2019
Week 5. Differential Equations, Models Growth proportional to size: Logistic equation: Force balance: gravitation: restoring force of a spring: friction: or or 5/11/2019
Week 5. Differential Equations, Direction Field The direction field without and with some solution curves. Example: y’ = x 2 + y 2 - 1 5/11/2019
Week 5. Separable Differential Equations Separable if: Solution steps: separate: with formal integration: find anti-derivatives: with fixed, arbitrary rewrite (if possible): 5/11/2019
Week 5. Differential Equations for Orthogonal Trajectories A given family of curves: The set of orthogonal trajectories is the family of ‘perpendicular’ curves recipe: rewrite original family to a d.e.: use the orthogonality condition solve: 5/11/2019
Week 5. First-order Linear Differential Equations First-order linear if: Solution steps: solve reduced equation: i.e. variation of constants, substitute and the d.e. becomes: ‘separate’ the previous d.e. and solve for , that is: with in rewrite: (a.k.a. variation of parameters) Note that the d.e. in and in are separable. 5/11/2019
Week 6. Structure of solutions to 1st-order Linear D.E. The d.e. without right hand side Q(x ) is called homogeneous. 5/11/2019
Week 6. Structure of solutions to 2nd-order Linear D.E. The d.e. without right hand side R(x ) is called homogeneous. Two functions are independent if y1(x ) + y2(x ) = 0 for all x with , two fixed numbers, implies = = 0. 5/11/2019
Week 6. Homogeneous 2nd-order Linear D. E Week 6. Homogeneous 2nd-order Linear D.E. with constant coefficients, a. 5/11/2019
Week 6. Homogeneous 2nd-order Linear D. E Week 6. Homogeneous 2nd-order Linear D.E. with constant coefficients, an example 5/11/2019
Week 6. Homogeneous 2nd-order Linear D. E Week 6. Homogeneous 2nd-order Linear D.E. with constant coefficients, b. 5/11/2019
Week 6. Homogeneous 2nd-order Linear D. E Week 6. Homogeneous 2nd-order Linear D.E. with constant coefficients, c. From complex to real Remember that the bar stands for complex conjugate. 5/11/2019
Week 6. Homogeneous 2nd-order Linear D. E Week 6. Homogeneous 2nd-order Linear D.E. with constant coefficients, an example 5/11/2019
Week 6. General 2nd-order Linear D.E. with constant coefficients This is a variation on the ‘method of variation of parameters’, page 1136. 5/11/2019
Week 6. Not so very general 2nd-order Linear D. E Week 6. Not so very general 2nd-order Linear D.E. with constant coefficients This is so-called ‘method of undetermined coefficients’, page 1132. One might call it ‘clever guessing’. For some p and q these guesses above are not clever enough…. 5/11/2019