1. Notes for Analysis Et/WiGS
TU Delft
2001
5/11/2019

2. Week 1. Complex numbers a.
5/11/2019

3. Week 1. Complex numbers b.
5/11/2019

4. Week 1. Complex numbers c.
5/11/2019

5. Week 1. Complex numbers d.
Only for n4 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

6. Week 2. Limit, the definition
5/11/2019

7. Week 2. Derivative, the definition
5/11/2019

8. Week 2. Application of Implicit Differentiation
5/11/2019

9. Week 2. Special functions
5/11/2019

10. Week 3. Mean Value Theorem a.
5/11/2019

11. Week 3. Mean Value Theorem b.
5/11/2019

12. Week 3. Antiderivatives and Integrals
Antiderivative  ‘inverse derivative’
Integral  ‘signed area under graph’
5/11/2019

13. 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

14. 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

15. Week 3. Substitution Rule
5/11/2019

16. Week 3. Partial Integration Rule
5/11/2019

17. Week 3. Both rules in shorthand
If one doesn’t know why, just magic remains.
5/11/2019

18. Week 3. Partial Integration, an example
Not all v’s are equal.
5/11/2019

19. Week 4. Integration of Rational Functions, a.
5/11/2019

20. Week 4. Integration of Rational Functions, b.
5/11/2019

21. 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

22. Week 4. Splitting of Rational Functions
5/11/2019
But what about complex roots?

23. 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

24. 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

25. Week 4. Integration of Rational Functions, an example
 Complex
Real 
5/11/2019

26. 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

27. 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

28. Week 4. Improper Integrals, b.
Unbounded interval
5/11/2019

29. Week 4. Improper Integrals, c.
Unbounded function :
5/11/2019

30. Week 4. Improper Integrals, Comparison Test
A similar comparison test can be formulated for improper integrals of the second type.
5/11/2019

31. Week 4. Special Improper Integrals
What about ?
The improper integrals above are often candidates for the comparison test.
5/11/2019

32. 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

33. 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

34. 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

35. 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

36. 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

37. 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

38. 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

39. 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

40. Week 6. Homogeneous 2nd-order Linear D. E
Week 6. Homogeneous 2nd-order Linear D.E. with constant coefficients, a.
5/11/2019

41. Week 6. Homogeneous 2nd-order Linear D. E
Week 6. Homogeneous 2nd-order Linear D.E. with constant coefficients, an example
5/11/2019

42. Week 6. Homogeneous 2nd-order Linear D. E
Week 6. Homogeneous 2nd-order Linear D.E. with constant coefficients, b.
5/11/2019

43. 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

44. Week 6. Homogeneous 2nd-order Linear D. E
Week 6. Homogeneous 2nd-order Linear D.E. with constant coefficients, an example
5/11/2019

45. 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

46. 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 One might call it ‘clever guessing’. For some p and q these guesses above are not clever enough….
5/11/2019