# Download A Course of Pure Mathematics 3rd ed by G. H. Hardy PDF

By G. H. Hardy

This ebook has been designed basically for using first yr scholars on the Universities whose skills succeed in or technique whatever like what's frequently defined as scholarship general. i'm hoping that it can be precious to different sessions of readers, however it is that this classification whose desires i've got thought of first. It os at the least ebook for mathematicians i've got nowhere made any try to meet the wishes of scholars of engineering or certainly any classification of scholars whose pursuits should not basically mathematical.

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**Example text**

If # 3 = x + 1 then #3n = a Hx + bn + cjx, where 38. If #G +^5 -2. r3 +tf 2 + l=0 a quadratic equation with rational [It will 37 be found that y 1 + y + 1 and y = A4 -^2 4-^- coefficients. ] 1, then (Afat/i. y satisfies Trip. ) CHAPTER II FUNCTIONS OF HEAL VARIABLES The idea of a function. Suppose that x and y are 20. two continuous real variables, which we may suppose to be repre sented geometrically by distances A^P = x, B^Q^y measured from fixed points A 0) B along two straight lines A, M. And let us suppose that the positions of the points P and Q are not independent, but connected by a relation which we can imagine to be expressed as a relation between x and y: so that, when P and x are known, Q and y are also known.

Express the polynomial (x say) in the form ly ... are rational, as in Ex. 22. 29 n, REAL VARIABLES 36 Similarly x2 =l2 + m 2 a+... +r 2 a (n Hence L x + L%x2 + where A v . . , + L nxn = A, ... r Z2 1) the determinant is mi and Zi, ~ the minors of ... , l lt 1 2 , 30. Apply this process to 31. ] x=p+Jq, and 3 1 + cp 2/3 satisfies deduce the theorem of 14. the equation 2 3 3 2 3 # - 3a/+ 3y (a - 6cjo) - a - 6 p - c ^ + Zabcp = 0. 3 32. We Algebraical numbers. have seen that some irrational numbers (such as x/2) are roots of equations of the type a^x \ } + a^xn ~ + n l .

The same is <Ja + *Jb true of *Ja cannot be rational unless Jb, unless a = b. Ja and \ REAL VARIABLES 14, 15] 23 If 4. then either (a) A = C and B=D, or (6) A =D and J3 = C,or (c) JA, *JB, ^<7, [Square the given equation and ^/^) are all rational or all similar surds. ] Neither (a + Jb) 3 nor (a - 5. 3 */b} can be rational unless Jb is rational. m where if x=p + *Jq, where p and q are rational, then # where P and Q the form in can be P+QJq, expressed any integer, are rational. For example, Prove that 6.