This textbook presents an algorithmic approach to mathematical analysis, with a focus on modelling and on the applications of analysis. Fully integrating mathematical software into the text as an important component of analysis, the book makes thorough use of examples and explanations using MATLAB, Maple, and Java applets. Mathematical theory is described alongside the basic concepts and methods of numerical analysis, supported by computer experiments and programming exercises, and an extensive use of figure illustrations. Features: thoroughly describes the essential concepts of analysis; provides summaries and exercises in each chapter, as well as computer experiments; discusses important applications and advanced topics; presents tools from vector and matrix algebra in the appendices, together with further information on continuity; includes definitions, propositions and examples throughout the text; supplementary software can be downloaded from the book’s webpage.

Example 4.3 (Taking the square root of complex numbers) The equation z2 = a + ib can be solved by the ansatz (x + iy)2 = a + ib so x 2 − y 2 = a, 2xy = b. If one uses the second equation to express y through x and substitutes this into the first equation, one obtains the quartic equation x 4 − ax 2 − b2 /4 = 0. Solving this by the substitution t = x 2 one obtains two real solutions. In the case of b = 0, either x or y equals zero depending on the sign of a. The Complex Plane A geometric

+ tan (arctan x) 1 + x 2 (arcsin x)′ = 1 2 −1 < x < 1, −1 < x < 1, −∞ < x < ∞. The derivatives of the most important elementary functions are collected in Table 7.1. The formulae are valid on the respective domains. 7.5 Numerical Differentiation In applications it often happens that a function can be evaluated for arbitrary arguments but no analytic formula is known which represents the function. This situation, for example, arises if the dependent variable is determined using a measuring

Parametric curves in space. 4 J.A. Lissajous, 1822–1880. 188 14 Curves 4. (a) Using the Java applet analyse where the cycloid x(t) = t − 2 sin t, − 2π ≤ t ≤ 2π y(t) = 1 − 2 cos t, has its maximal speed (∥˙x(t)∥ → max), and check your result by hand. (b) Discuss and explain the shape of the loops x(t) = cos nt cos t, 0 ≤ t ≤ 2π y(t) = cos nt sin t, for n = 1, 2, 3, 4, 5, using the Java applets (plot the moving frame). 5. Study the velocity and the acceleration of the following curves

Functions Fig. 2.1 A function a row vector of the same length of corresponding y-values is generated. Finally, plot(x,y) plots the points (x1 , y1 ), . . . , (xn , yn ) in the coordinate plane and connects them with line segments. The result can be seen in Fig. 2.1. In the general mathematical framework we do not just want to assign finite lists of values. In many areas of mathematics functions defined on arbitrary sets are needed. For the general set-theoretic notion of a function we refer to

algorithmic approach chosen by us encompasses: (a) Development of concepts of analysis from an algorithmic point of view. (b) Illustrations and explanations using M ATLAB and maple programs as well as Java applets. (c) Computer experiments and programming exercises as motivation for actively acquiring the subject matter. (d) Mathematical theory combined with basic concepts and methods of numerical analysis. Concise presentation means for us that we have deliberately reduced the subject matter to