The History of Mathematics: A Brief Course

(^482) 16. THE CALCULUS 9 = 2 \j = 3+arctan fcx/ arctan fc J/ = 4 FIGURE 7. The functional $(y) = J_ 1 (xy'(x)) dx does not as- ...

BRANCHES AND ROOTS OF THE CALCULUS 483 a minimum assuming that all the relevant quantities are differentiable a sufficient num ...

484 16. THE CALCULUS Perhaps it may be objected, that there is no ultimate proportion of evanescent quantities; because the prop ...

BRANCHES AND ROOTS OF THE CALCULUS 485 The foreign mathematicians are supposed by some, even of our own, to proceed in a manne ...

486 16. THE CALCULUS distance traveled in an instant of time by an object moving with finite velocity. This debate was carried o ...

QUESTIONS AND PROBLEMS 487 equivalent to infinitesimals. He defined a function f(x) to be continuous if the absolute value of th ...

(^488) 16. THE CALCULUS 16.3. Show that the point at which the tangent to the curve y = f(x) intersects the y axis is y = f(x) - ...

Chapter 17. Real and Complex Analysis In the mid-1960s Walter Rudin (b. 1921), the author of a number of standard grad- uate tex ...

490 17. REAL AND COMPLEX ANALYSIS complex variable whose range was much vaster than the materials that led to its creation. In h ...

COMPLEX ANALYSIS 491 for rectification and an equation fy 1 1 fx 1 / , dt=- „ dt, Jo %/T^t^2 Wo which can be written in differ ...

492 17. REAL AND COMPLEX ANALYSIS Legendre, Jacobi, and Abel. The foundation for further work in integration was laid by Legendr ...

COMPLEX ANALYSIS 493 inverses. For example, if q(x) is of degree 5, he posed the problem of solving for ÷ and y in terms of u ...

494 17. REAL AND COMPLEX ANALYSIS l/(x + iy) did not have this property if the rectangle contained the point (0,0). The differen ...

COMPLEX ANALYSIS 495 counted according to order, less the number of sheets in the surface, plus l.^3 The Riemann surface of w ...

496 17. REAL AND COMPLEX ANALYSIS 2. Real analysis In complex analysis attention is restricted from the outset to functions that ...

REAL ANALYSIS 497 For a string fastened at two points, say (0,0) and (L, 0) and vibrating so that its displacement above or be ...

498 17. REAL AND COMPLEX ANALYSIS of gravity, Pierre-Simon Laplace (1749-1827) was led to what is now known as Laplace's equatio ...

REAL ANALYSIS 499 Cauchy's definition still does not specify what ways of expressing one variable in terms of another are legi ...

500 17. REAL AND COMPLEX ANALYSIS is called the Dirichlet function. For such a function, he thought, no integral could be define ...

REAL ANALYSIS 501 [They] seem in general to be limited to a relatively narrow condi- tion, one which is insufficient for even ...