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FUNCTIONS

#### 35.2.1 Discrete and Continuous Probability Distributions

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Figure 35.9 Distribution functions

#### 35.2.2 Basic Polynomials

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• The quadratic equation appears in almost every engineering discipline, therefore is of great importance.

Figure 35.10 Quadratic equation

• Cubic equations also appear on a regular basic, and as a result should also be considered.

Figure 35.11 Cubic equations

• On a few occasions a quartic equation will also have to be solved. This can be done by first reducing the equation to a quadratic,

Figure 35.12 Quartic equations

#### 35.2.3 Partial Fractions

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• The next is a flowchart for partial fraction expansions.

Figure 35.13 The methodolgy for solving partial fractions

• The partial fraction expansion for,

Figure 35.14 A partial fraction example

• Consider the example below where the order of the numerator is larger than the denominator.

Figure 35.15 Solving partial fractions when the numerator order is greater than the denominator

• When the order of the denominator terms is greater than 1 it requires an expanded partial fraction form, as shown below.

Figure 35.16 Partial fractions with repeated roots

• We can solve the previous problem using the algebra technique.

Figure 35.17 An algebra solution to partial fractions

#### 35.2.4 Summation and Series

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• The notation
is equivalent to
assuming
and
are integers and
. The index variable
is a placeholder whose name does not matter.

• Operations on summations:

• Some common summations:

for both real and complex
.

for both real and complex
. For
, the summation does not converge.

#### 35.2.5 Practice Problems

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1. Convert the following polynomials to multiplied terms as shown in the example.

2. Solve the following equation to find ‘x’.

3. Reduce the following expression to partial fraction form.

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