tailieunhanh - Ebook Engineering mathematics (5/E): Part 2

Part 2 book “Engineering mathematics” has contents: Complex numbers, presentation of statistical data, measures of central tendency and dispersion, the binomial and poisson distribution, the normal distribution, differentiation of implicit functions, differentiation of implicit functions, integration using algebraic substitutions, and other contents. | Section 6 Complex Numbers This page intentionally left blank Chapter 35 Complex numbers Cartesian complex numbers (i) If the quadratic equation x 2 + 2x + 5 = 0 is solved using the quadratic formula then: (2)2 − (4)(1)(5) x= 2(1) √ √ −2 ± −16 −2 ± (16)(−1) = = 2 2 √ √ √ −2 ± 16 −1 −2 ± 4 −1 = = 2 2 √ = −1 ± 2 −1 √ It is not possible to evaluate −1 in real terms. √ However, if an operator j is defined as j = −1 then the solution may be expressed as x = −1 ± j2. −2 ± √ Since x 2 + 4 = 0 then x 2 = −4 and x = −4 ., (Note that ±j2 may also be written as ± 2j). Problem 2. Problem 1. Solve the quadratic equation: x2 + 4 = 0 Solve the quadratic equation: 2x 2 + 3x + 5 = 0 Using the quadratic formula, −3 ± (3)2 − 4(2)(5) 2(2) √ √ √ −3 ± −31 −3 ± −1 31 = = 4 4 √ −3 ± j 31 = 4 x= (ii) −1 + j2 and −1 − j2 are known as complex numbers. Both solutions are of the form a + jb, ‘a’ being termed the real part and jb the imaginary part. A complex number of the form a + jb is called a Cartesian complex number. (iii) In pure √ mathematics the symbol i is used to indicate −1 (i being the first letter of the word imaginary). However i is the symbol of electric current in engineering, and to avoid possible confusion the√ next letter in the alphabet, j, is used to represent −1 √ √ (−1)(4) = −1 4 = j(±2) √ = ± j2, (since j = −1) x= Hence √ 31 3 or − ± , x=− +j 4 4 correct to 3 decimal places. (Note, a graph of y = 2x 2 + 3x + 5 does not cross the x-axis and hence 2x 2 + 3x + 5 = 0 has no real roots). Problem 3. (a) j3 Evaluate (b) j4 (c) j23 (d) −4 j9 314 Engineering Mathematics (a) j3 = j2 × j = (−1) × j = −j, since j2 = −1 (b) j4 = j2 × j2 = (−1) × (−1) = 1 Imaginary axis B (c) j23 = j × j22 = j × ( j2 )11 = j × (−1)11 j4 j3 = j × (−1) = −j (d) j =j×1=j Hence A j2 j9 = j × j8 = j × ( j2 )4 = j × (−1)4 3 2 1 −4 −4 −4 −j 4j = = × = 2 j9 j j −j −j 4j = = 4 j or j4 −(−1) 0 j 1 2 3 Real axis j 2 j 3 D j 4 Now

• Toán học
• Engineering mathematics
• Complex numbers
• Central tendency
• Poisson distribution
• The normal distribution
• Integration using algebraic substitutions
• Advanced engineering mathematics
• Advanced engineering
• Advanced mathematics
• Ordinary differential equations
• Power series solutions
• The eigenvalue problem
• Vectors in 3 space
• First Order ODEs
• Second Order linear ODEs
• Qualitative methods
• Laplace transforms
• Complex integration
• Numeric linear algebra
• Potential theory
• Complex analysis
• Differential equations
• The laplace transform
• Series solutions
• Approximation of solutions
• Fourier series
• The fourier integral
• The potential equation
• The residue theorem
• Higher engineering mathematics
• Hyperbolic functions
• Geometric progressions
• Partial fractions
• The binomial series
• Parametric equations
• Implicit functions
• Integration using partial fractions
• Some applications of integration
• Engineering notation
• Computer numbering systems
• Further algebra
• Simple equations
• Simultaneous equations
• Basic engineering mathematics
• Basic arithmetic
• Quadratic equations
• Higher order linear ODEs
• Vector differential calculus
• Fourier analysis
• Partial differential equations
• Laurent series
• Residue integration
• Conformal mapping
• Mathematics and engineering
• Complex functions
• Harmonic functions
• Mandelbrot sets
• Elementary functions
• Residue theory
• Laplace transform
• Systems of ODEs
• Phase plane
• Numerics in general
• Introduction to trigonometry
• Trigonometric waveforms
• Areas of plane figures
• Volumes of common solids
• Adding of waveforms
• Journal of technology and science education
• Post secondary project based learning in science
• Post secondary project based learning in technology
• Post secondary project based learning in engineering and mathematics
• Post secondary project
• Mathematics as profession
• Definitions of mathematics
• Quantity
• Statistics and other decision sciences
• Mechanical engineers pocket book
• Engineering statics
• Engineering dynamics
• Orders of levers
• Stress and strain
• Screwed fastenings
• Riveted joints
• Mathematics for Finance
• An Introduction
• to Financial Engineering
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• Blackledge
• P
• Yardley Applied Geometry
• Mechanical engineers reference
• Alternative energy sources
• Nuclear engineering
• Offshore engineering
• Plant engineering
• Manufacturing methods