advanced mathematical methods for scientists and engineers i pdf

... look for books with “Introduction” or “Elementary” in the title. If it is an “Intermediate” text it will be incomprehensible. If it is Advanced then not only will it be incomprehensible, it will have ... . 2039 xx Injective Surjective Bijective Figure 1.1: Depictions of Injective, Surjective and Bijective Functions 1.3 Inverses and Multi-Valued Functions If y = f(x), then we can write x = f −1 (y) ... words, distinct elements are mapped to distinct elements. f is surjective if for each y in the codomain, there is an x such that y = f(x). If a function is both injective and surjective, then it is...

Ngày tải lên: 06/08/2014, 01:21

... Delta and Einstein Summation Convention The Kronecker Delta tensor is defined δ ij =  1 if i = j, 0 if i = j. This notation will be useful in our work with vectors. Consider writing a vector in ... (b), associativity of scalar multiplication. ã a à (b + c) = a · b + a · c, distributive. (See Exercise 2.1.) ã e i e j = ij . In three dimensions, this is i · i = j · j = k · k = 1, i · j = j ... lim x→ξ y(x) = 0. Left and Right Limits. With the notation lim x→ξ + y(x) we denote the right limit of y(x). This is the limit as x approaches ξ from above. Mathematically: lim x→ξ + exists if...

Ngày tải lên: 06/08/2014, 01:21

... and right limits do exist, then the function has a finite discontinuity. If either the left or right limit does not exist then the function has an infinite discontinuity. 76 Relative Extrema and ... polynomial approximation of this function near the 71 Figure 3.4: Piecewise Continuous Functions the function is said to be uniformly continuous on the interval. A sufficient cond ition for uniform ... thus discontinuous at that point. Since the numerator and denominator are continuous functions and the 75 Figure 3.3: A Removable discontinuity, a Jump Discontinuity and an Infinite Discontinuity Boundedness....

Ngày tải lên: 06/08/2014, 01:21

... 1)! f (n+1) (ξ). Solution 3.7 Consider lim x→0 (sin x) sin x . This is an indeterminate of the form 0 0 . The limit of the logarithm of the expression is lim x→0 sin x ln(sin x). This is an indeterminate of the form ... expression to obtain an indeterminate of the form ∞ ∞ and then apply L’Hospital’s rule. lim x→0 ln(sin x) 1/ sin x = lim x→0 cos x/ sin x −cos x/ sin 2 x = lim x→0 (−sin x) = 0 The original limit is lim x→0 (sin ... if it is differentiable, it has an infinite number of indefinite integrals, each of which differ by an additive constant. Zero Slope Implies a Constant Function. If the value of a function’s derivative...

Ngày tải lên: 06/08/2014, 01:21

... is r  = − 1 4 cos  t 2  i − 1 4 sin  t 2  j. See Figure 5.8 for plots of position, ve locity and acceleration. Figure 5.8: A Graph of Position and Velocity and of Position and Acceleration Solution ... Suppose you are standing on some terrain. The slope of the ground in a particular direction is the directional derivative of the elevation in that direction. Consider a differentiable scalar field, ... a function of time. The function is continous at a point t = τ if lim t→τ r(t) = r(τ). This occurs if and only if the component functions are continu ous. The function is differentiable i f dr dt ≡...

Ngày tải lên: 06/08/2014, 01:21

... arctan(x,y) . Cartesian form is convenient for addition. Polar form is convenient for multiplication and division. Example 6.3.1 We write 5 + ı7 in polar form. 5 + ı7 = √ 74 e ı arctan(5,7) We write 2 e ıπ/6 in ... trigonometric functions with some fairly messy trigonometric identities. This would take much more work than directly multiplying (5 + ı7) 11 . 6.6 Rational Exponents In this section we consider ... definition of exponentiation, we have e ınθ =  e ıθ  n We apply Euler’s formula to obtain a result which is useful in deriving trigonometric identities. cos(nθ) + ı sin(nθ) = (cos θ + ı sin θ) n Result...

Ngày tải lên: 06/08/2014, 01:21

... direction. For circles, the positive direction is the counter-clockwise direction. The positive direction is consistent with the way angles are measured in a right-handed coordinate system, i. e. ... the positive imaginary axis and approach infinity via pure imaginary numbers. We could generalize the real variable notion of signed infinity to a complex variable notion of directional infinity, ... difference in 223 Figure 7.3: Traversing the boundary in the positive direction. Two interpretations of a curve. Consider a simple closed curve as depicted in Figure 7.4a. By giving it an orientation,...

Ngày tải lên: 06/08/2014, 01:21

... the familiar definitions in terms of the exponential function. Thus not surprisingly, we can write the sine in terms of the hyperbolic sine and write the cosine in terms of the hyperbolic cosine. ... at infinity and its only singularity is at z = 1, the only possi bili ties for branch points are at z = 1 and z = ∞. Since log  1 z −1  = −log(z −1) and log w has branch points at zero and infinity, ... the real and imaginary parts of the cosine and sine, respectively. Figure 7.20 shows the modulus of the cosine and the sine. The hyperbolic sine and cosine. The hyperbolic sine and cosine have...

Ngày tải lên: 06/08/2014, 01:21

... modulus-argument form. Hint 7.4 Write e z in polar form. Hint 7.5 The exponential is an increasing function for real variables. Hint 7.6 Write the hyperbolic cotangent in terms of exponentials. Hint 7.7 Write ... verify this solution. 1 z = e z log(1) = e ız2πn For n = 0, this has the value 1. Logarithmic Identities Solution 7.9 We write the relationship in terms of the natural logarithm and the principal ... positive real axis with an accumulation point at the origin. See Figure 7.40. 313 is defined on the positive real axis. Define a branch such that f(1) = 1/ 3 √ 2. Write down an explicit formula for...

Ngày tải lên: 06/08/2014, 01:21

... derivative in terms of the derivative in a coordinate direction. However, we don’t have a nice way of determining if a function is analytic. The definition of complex d erivative in terms of a limit is ... this to obtain two equations.) A sufficient condition for analyticity of f(z) is that the Cauchy-Riemann equations hold and the first partial derivatives of φ exist and are continuous in a neighborhood of ... exponential. In Exercise 8.13 you can sh ow that the logarithm log z is differentiable for z = 0. This implies the differentiability of z α and the inverse trigonometric functions as they can be written...

Ngày tải lên: 06/08/2014, 01:21

... Cauchy-Riemann equations for à and are satised if and only if the Cauchy-Riemann equations for u and v are satisfied. The continuity of the first partial derivatives of u and v implies the same of à and ... essential si ngularity is, we say what it is not. If z 0 neither a branch point, a removable singularity nor a pole, it is an essen tial singularity. A pole may be called a non-essential singularity. ... z 0 ) n f(z) is analytic there. 8.4.2 Isolated and Non-Isolated Singularities Result 8.4.3 Isolated and Non-Isolated Singularities. Suppose f(z) has a singularity at z 0 . If there exists a deleted neighb...

Ngày tải lên: 06/08/2014, 01:21

... hyperbolic sine. Since the hyperbolic sine has an essential singularity at infinity, the function has an essential singularity at i nfini ty as well. The point at infinity is a non-isolated si ngularity ... line y = x−1 and the partial derivatives are continuous, the function f(z) is differentiable there. Since the function is not differentiable in a neighborhood of any point, it is nowhere analytic. 423 ... the partial derivatives are continuous, f(z) is everywhere differentiable. Since f(z) is differentiable in a neighborhood of every point, it is analytic in the complex plane. (f(z) is entire.) Now...

Ngày tải lên: 06/08/2014, 01:21

... contour with the opposite orientation. Let 469 This function is analytic where f(ζ) is analytic. It is a simple calculus exercise to show that the complex derivative in the ξ direction, ∂ ∂ξ , and ... two evils, I always pick the one I never tried before. - Mae West 10.1 Line Integrals In this section we will recall the definition of a line integral in the Cartesian plane. In the next section ... will use this to define the contour integral in the complex plane. Limit Sum Definition. First we develop a limit sum definition of a line integral. Consider a curve C in the Cartesian plane joining...

Ngày tải lên: 06/08/2014, 01:21

... 0 arg(sin(z))   C = 2π Solution 11.2 1. Since the integrand sin z z 2 +5 is analytic inside and on the contour, (the only singularities are at z = ±ı √ 5 and at infinity), the integral is zero ... 1) dz 495 If the limit is greater than unity, then the terms are eventually increasing with n. Since the terms do not vanish, the sum is divergent. If the limit is less than unity, then there exists ... 1 where g is analytic inside and on C, (the positive circle |z| = 1), then  C f(z) dz = ı2πα 1 . Exercise 11.8 Show that if f(z) is analytic within and on a simple closed contour C and z 0 is not...

Ngày tải lên: 06/08/2014, 01:21

... −ζ| < δ in the domain. An equivalent definition is that f(z) is continuous in a closed domain if lim ζ→z f(ζ) = f(z) for all z in the domain. Convergence. Consider a series in which the terms ... jump discontinuities at x = 2kπ and is continuous on any closed interval not containing one of those points. 12.3 Uniformly Convergent Power Series Power Series. Power series are series of the form ∞  n=0 a n (z ... − N  n=1 a n (z)      <  for all z in the domain. 12.2.1 Tests for Uniform Convergence Weierstrass M-test. The Weierstrass M-test is useful in determining if a series is uniformly convergent. The series  ∞ n=0 a n (z)...

Ngày tải lên: 06/08/2014, 01:21

Ngày tải lên: 06/08/2014, 01:21

Ngày tải lên: 06/08/2014, 01:21

Ngày tải lên: 06/08/2014, 01:21

Ngày tải lên: 06/08/2014, 01:21

Ngày tải lên: 06/08/2014, 01:21