# An Introduction to the Fractional Calculus and Fractional by Kenneth S. Miller

By Kenneth S. Miller

Commences with the ancient improvement of fractional calculus, its mathematical theory—particularly the Riemann-Liouville model. quite a few examples and theoretical functions of the idea are provided. gains themes linked to fractional differential equations. Discusses Weyl fractional calculus and a few of its makes use of. comprises chosen actual difficulties which result in fractional differential or necessary equations.

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**Additional resources for An Introduction to the Fractional Calculus and Fractional Differential Equations**

**Example text**

Write with common denominator. Add fractions. Simplify. (b) In this case, the least common denominator is 4x͑x ϩ 2͒. 1 1 1 1 ϭ Ϫ Ϫ 2͑x 2 ϩ 2x͒ 4x 2x͑x ϩ 2͒ 2͑2x͒ 2 xϩ2 Ϫ 2͑2x͒͑x ϩ 2͒ 2͑2x͒͑x ϩ 2͒ 2ϪxϪ2 ϭ 4x͑x ϩ 2͒ ϭ TRY IT ϭ Ϫx 4x͑x ϩ 2͒ ϭ Ϫ1 , 4͑x ϩ 2͒ Factor. Write with common denominator. Subtract fractions. Divide out like factor. x 0 Simplify. 2 Perform each indicated operation and simplify. (a) x 2 ϩ x2 Ϫ 4 x Ϫ 2 ALGEBRA (b) 1 1 Ϫ 3͑x2 ϩ 2x͒ 3x REVIEW To add more than two fractions, you must find a denominator that is common to all the fractions.

Example 3 shows how an inequality can be used to describe the production level of a manufacturing plant. 50 per unit. During the month of August, the total cost of production varied from a high of $1325 to a low of $1200 per day. Find the high and low production levels during the month. 5x to produce x units. 5x ϩ 500. Now, because the cost ranged from $1200 to $1325, you can write the following. 5x ϩ 500 Write original inequality. 5x ϩ 500 Ϫ 500 ≤ 1325 Ϫ 500 Subtract 500 from each side. 5x Simplify.

Rational Zero Theorem If a polynomial an x n ϩ a nϪ1 x nϪ1 ϩ . . ϩ a1 x ϩ a 0 has integer coefficients, then every rational zero is of the form x ϭ p͞q, where p is a factor of a 0, and q is a factor of a n. EXAMPLE 4 Using the Rational Zero Theorem Find all real zeros of the polynomial. 2x 3 ϩ 3x 2 Ϫ 8x ϩ 3 SOLUTION 2 x 3 ϩ 3x 2 Ϫ 8x ϩ 3 Factors of constant term: ± 1, ± 3 STUDY Factors of leading coefficient: ± 1, ± 2 The possible rational zeros are the factors of the constant term divided by the factors of the leading coefficient.