Ceiling. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation. Specialized values

Transcription

1 Ceiling Notations Traditional name Ceiling function Traditional notation z Mathematica StandardForm notation Ceilingz Primary definition x n ; x n n 1 x n z Rez Imz For real z, the function z is the smallest integer greater than or equal to z. Examples: 3. 4, 3 3, 0. 0,.3, 3 1, Π 3, , 5 3, 7 4. Specific values Specialized values x x ; x x x ; x x y x y ; x y Values at fixed points

2 Π Values at infinities General characteristics Domain and analyticity z is a nonanalytical function; it is a piecewise constant function which is defined over the whole complex z-plane zz

3 3 Symmetries and periodicities Mirror symmetry z z 1 Χ Imz Periodicity No periodicity Sets of discontinuity The function z is a piecewise constant function with unit jumps on the lines Rez k Imz l ; k, l. The function z is continuous from the left on the intervals k, k, k, and from below on the intervals k, k, k z z k, k, 1 ; k, k, k, ; k lim z Ε z ; Rez Ε lim z Ε z 1 ; Rez Ε lim z Ε z ; Imz Ε lim z Ε z ; Imz Ε0 Series representations Exponential Fourier series x x 1 1 Π sin Π k x ; x x k 1 k Other series representations m n m n 1 n1 n sin Π k m n k 1 cot Π k n 1 ; m n m n n 1 Transformations Transformations and argument simplifications

4 4 Argument involving basic arithmetic operations z z ; Rez Imz z z sgnimz sgnrez ; Rez Imz z z 1 Χ Imz sgnimz 1 Χ Rez sgnrez z z Χ Imz z Imz Rez z z 1 Χ Rez z Imz Rez z n z n ; n n x n x ; x n Argument involving related functions z z z z z z z z intz intz fracz 1 Χ Imz ΘImz 1 Χ Rez ΘRez quotientm, n m n Nest f, x, n x a b b 1 bn a b b 1 ; f x a x b a b Addition formulas

5 z n z n ; n z 1 z z 1 z z 1 z z 1 z Multiple arguments n1 kn k 1 n z n z k Θ z mod Θ z mod 1 1 n k 0 ; n z Products, sums, and powers of the direct function Sums of the direct function z 1 z z 1 z z 1 z z 1 z n1 k 0 x k m n m1 k 0 x k n m ; x n m Complex characteristics Real part Rex y x Rez Rez Imaginary part Imx y y Imz Imz Absolute value x y x y z Imz Rez Argument argx y tan 1 x, y

6 argz tan 1 Rez, Imz Conjugate value x y x y z Rez Imz Signum value x y sgnx y x y sgnz z z Differentiation Low-order differentiation z z x x x k k In a distributional sense, for x. Fractional integro-differentiation Α z z Α z zα 1 Α Integration Indefinite integration Involving only one direct function z z z z Involving one direct function and elementary functions

7 7 Involving power function z Α1 z z zα z Α z z logz z z Definite integration For the direct function itself In the following formulas a n n n 1 tt ; n att 1 a a 1 a a t Α1 tt a aα ΖΑ ΖΑ, a ; ReΑ 0 0 Α a t Α1 tt 1 Α a aα ΖΑ, a ; ReΑ ΖΑ 1 t Α1 tt ; ReΑ 1 1 Α a att a Integral transforms Fourier exp transforms t tz Π z Π k Π z Π k z k k 1 Π z Fourier cos transforms

8 c t tz Π z 1 cot z Π z Fourier sin transforms s t tz Π z 1 Π z Π k Π z Π k z k k 1 Laplace transforms z t tz ; Rez 0 z 1 z Summation Finite summation y k 0 x k y x y x 1 y y ; x y 0 x 1 0 y 1 n1 1 k 0 p k m p m n m n m p m 1 p n m ; n m p Representations through equivalent functions With related functions With Floor For real arguments x x 1 ; x x x x ; x x x ΘΧ x 1 1 ; x For complex arguments z z ; Rez Imz

9 z z 1 ; Rez Imz z z ; Rez Imz z z 1 ; Rez Imz z z ΘΧ Rez 1 Θ Χ Imz z z With Round For real arguments x x 1 x 1 ; x x x 1 x 1 1 ; x x 1 Χ x 1 ; x For complex arguments z Rez 1 z Χ Χ Imz 1 With IntegerPart For real arguments x intx 1 ; x x 0 x x intx ; x x 0 x x intx sgnχ x Θx 1 ; x For complex arguments z intz 1 sgnχ Rez ΘRez sgnχ Imz ΘImz

10 10 With FractionalPart For real arguments x x fracx 1 ; x x 0 x x x fracx ; x x 0 x x x fracx sgnχ x Θx 1 ; x For complex arguments z z fracz 1 sgnχ Rez ΘRez sgnχ Imz ΘImz With Mod z z z mod 1 With Quotient z quotientz, 1 With elementary functions z z tan1 cotπ z 1 ; z z Π Zeros z 0 ; 1 Rez 0 1 Imz 0 History C. F. Gauss (1808) K. E. Iverson (196) suggested the notation z

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