Notes for Analysis Et/Wi

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Notes for Analysis Et/Wi

• GS
• TU Delft
• 2001

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Week 1. Complex numbers a.
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Week 1. Complex numbers b.
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Week 1. Complex numbers c.
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Week 1. Complex numbers d.

• Only for n?4 the roots can be computed
  algebraically.
• For n2 by the abc-formula.
• For n3 by Cardanos method.
• For n4 by Ferraris method.

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Week 2. Limit, the definition
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Week 2. Derivative, the definition
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Week 2. Application of Implicit Differentiation
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Week 2. Special functions
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Week 3. Mean Value Theorem a.
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Week 3. Mean Value Theorem b.
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Week 3. Antiderivatives and Integrals

• Antiderivative ? inverse derivative
• Integral ? signed area under graph

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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.
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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

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Week 3. Substitution Rule
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Week 3. Partial Integration Rule
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Week 3. Both rules in shorthand
If one doesnt know why, just magic remains.
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Week 3. Partial Integration,an example
Not all vs are equal.
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Week 4. Integration of Rational Functions, a.
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Week 4. Integration of Rational Functions, b.
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Week 4. Division inRational Functions
Hence we only have to consider rational functions
with the denominator of lower degree than the
enumerator.
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Week 4. Splitting of Rational Functions
But what about complex roots?
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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.
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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.

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Week 4. Integration of Rational Functions, an
example
? Complex
Real ?
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Week 4. Integration of Rational Functions,
another example
After dividing out, finding roots, splitting
fractions, computing constants, combining,
substitution, one finally may integrate.
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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.

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Week 4. Improper Integrals, b.

• Unbounded interval

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Week 4. Improper Integrals, c.

• Unbounded function

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Week 4. Improper Integrals, Comparison Test
A similar comparison test can be formulated for
improper integrals of the second type.
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Week 4. Special Improper Integrals
What about ?
The improper integrals above are often candidates
for the comparison test.
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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.

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Week 5. Differential Equations, Models

• Growth proportional to size
• Logistic equation
• Force balance

• gravitation
• restoring force of a spring
• friction or
  or

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Week 5. Differential Equations, Direction Field
The direction field without and with some
solution curves.

• Example y x 2 y 2 - 1

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Week 5. Separable Differential Equations

• Separable if

• Solution steps
• separate
  with
• formal integration
• find anti-derivatives
  with fixed, arbitrary
• rewrite (if possible)

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Week 5. Differential Equations forOrthogonal
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

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Week 5. First-orderLinear 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.
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Week 6. Structure of solutions to 1st-order
Linear D.E.
The d.e. without right hand side Q(x ) is called
homogeneous.
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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.

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Week 6. Homogeneous 2nd-order Linear D.E. with
constant coefficients, a.
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Week 6. Homogeneous 2nd-order Linear D.E. with
constant coefficients, an example
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Week 6. Homogeneous 2nd-order Linear D.E. with
constant coefficients, b.
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Week 6. Homogeneous 2nd-order Linear D.E. with
constant coefficients, c.
From complex to real
Remember that the bar stands for complex
conjugate.
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Week 6. Homogeneous 2nd-order Linear D.E. with
constant coefficients, an example
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Week 6. General 2nd-order Linear D.E. with
constant coefficients
This is a variation on the method of variation
of parameters, page 1136.
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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.