Download Higher transcendental functions 2 by Arthur Erdelyi PDF

By Arthur Erdelyi

The Bulletin of the London Mathematical Society hailed this three-volume sequence as "The most generally pointed out mathematical works of all time and a simple reference resource for generations of utilized mathematicians and physicists in the course of the international. operating from huge notes on accepted targeted capabilities via the well known mathematician Harry Bateman, a group of editors not just comprehensive Bateman's unique venture but in addition made major advances in mathematical research. The books, which are used independently of one another, encompass quantity 1, which specializes in hypergeometric sequence; quantity 2, an exploration of Bessel features, orthogonal polynomials, and elliptic features and integrals; and quantity three, an exam of automorphic capabilities, spheroidal and ellipsoidal wave features, and different features.

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8 macht die Bezeichnung zweizeiliges HornerSchema verständlich. 00. 52 - 8 . 3) . 60i. 41. 3. 68i. Ein Paar von Nullstellen liegt offenbar vor, wenn der zugehörige Quadratfaktor das Ausgangspolynom ohne Rest teilt, das heißt, wenn r(x) in (34) verschwindet, also bN- 1 = b~ = bN = gilt. Aus (35) ist abzulesen, daß bN- 1 und bN Funktionen von ß und y sind. Im allgemeinen werden sie für die gewählten Startwerte noch nicht verschwinden. Man sucht deshalb nach Korrekturen h, k, für die ° bN-1(ß+h,y+k) =0, bN(ß + h, y + k) = ° (38) wird.

L (2) V + Ist ~ : = u i . v, i : = -1, eine komplexe Nullstelle des reellen Polynoms (1), so gehört auch die konjugiert komplexe Größe ~ : = u - i . v zu den Nullstellen. Durch Multiplikation der beiden Linearfaktoren x - ~ und x - ~ erhält man ~inen reellen Quadratfaktor x2 + ß· x + y:= (x -~). (x - ß:= -2Re ~ =-2u, ~), y : = (Re W + (Im ~)2 = u + v 2 (3) 2• Denkt man sich je zwei zusammengehörige komplexe Linearfaktoren in dieser Weise miteinander multipliziert und faßt man auch noch die Linearfaktoren von je zwei reellen Nullstellen zu einem Quadratfaktor zusammen, so erhält man für PN die stets reelle Produktdarstellung PN(X) = ao' Nj2 TI k=l (x 2 + ßkX + Yk) .

Praktisch nutzt diese Erkenntnis wenig, da im allgemeinen vor Beginn der Rechnung kaum Informationen über die Vielfachheit der Nullstelle vorliegen. Man kann sie sich aber während der Rechnung (zumindest näherungsweise) verschaffen. Dazu führen wir die Funktion F(x) := f(x)//,(x) ein. Sie besitzt auch im Fall mehrfacher Nullstellen von f bei x = x* nur eine einfache Nullstelle, denn wegen (69) ergibt sich F'(x*) = lim (1 _ x--+x. I(x~ I"~X») = t (x) 1/8 =1= O. 5. Bemerkungen Das auf F(x) angewandte Newton· Verfahren konvergiert demzufolge quadratisch.

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