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

Transcription

1 Round Notations Traditional name Nearest integer function Traditional notation z Mathematica StandardForm notation Roundz Primary definition x n ; x n x n z Rez Imz n 1 n ; n n n 1 n 1 ; For real z, the function x is the integer closest to z (if z ± 1, ± 3, ). Examples: 3. 3, 3 3, 0. 0,.3, 3 1, Π 3, 5 3, 5, 7. Specific values Specialized values x x ; x

2 x x ; x x y x y ; x y Values at fixed points Π Values at infinities

3 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 Symmetries and periodicities Parity z is an odd function z z Mirror symmetry z z Periodicity No periodicity Sets of discontinuity The function z is a piecewise constant function with unit jumps on the lines Rez 1 k Imz 1 l ; k, l. The function z is continuous from the right on the intervals k 1, k 1, k, and from the left on the intervals k 1, k 1, k. The function z is continuous from above on the intervals k, k, k, and from below on the intervals k, k, k z z k 1, k 1, 1 ; k, k 1, k 1, 1 ; k, k, k, ; k, k, k, ; k

4 lim z Ε z ; 1 Rez 1 Ε lim z Ε z ; 1 Rez 1 Ε lim z Ε z 1 ; 1 Re z 1 Ε lim z Ε z 1 ; 1 Rez 1 Ε lim z Ε z ; 1 Imz 1 Ε lim z Ε z ; 1 Imz 1 Ε lim z Ε z ; 1 Imz 1 Ε lim z Ε z ; 1 Imz 1 Ε0 Series representations Exponential Fourier series x x 1 Π 1 k sin Π k x ; x x 1 k 1 k Other series representations n m n m n 1 n1 sin Π k m n n k 1 cot Π k n 1 ; m n m n n 1 Transformations Transformations and argument simplifications Argument involving basic arithmetic operations

5 z z z z z z z n z n ; n n 1 1 z z 1 Argument involving related functions z z z z z z z z intz intz quotientm, n m n Addition formulas z n z n ; n Rez 1 Imz 1 Complex characteristics Real part Rex y x Rez Rez Imaginary part Imx y y Imz Imz

6 6 Absolute value x y x y z Rez Imz Argument argx y tan 1 x, y argz tan 1 Rez, Imz Conjugate value x y x y z Rez Imz Signum value x y sgnx y sgnz z z x y Differentiation Low-order differentiation z z 0 In a distributional sense for x x k x 1 x k Fractional integro-differentiation

7 7 Α z z Α zzα 1 Α Integration Indefinite integration Involving only one direct function z z z z Involving one direct function and elementary functions 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. 0 n tt n ; n att 1 a a a a t Α1 tt 1 Α a 1 aα a 1 1 Α 1 Α ΖΑ Ζ Α, a a t Α1 tt 1 Α a 1 aα Ζ Α, a 1 1 ; ReΑ t Α1 tt 1 Α Ζ Α, 3 1 ; ReΑ 1

8 t Α1 tt 1 Α Ζ Α, 3 Α ; ReΑ 1 a att Integral transforms Fourier exp transforms t tz 1 k k Π z Π k z Π k 1 k Π z Fourier cos transforms c t tz csc z Π z Fourier sin transforms s t tz 1 1 k k Π z Π k z Π k 1 k Π z Laplace transforms z t tz ; Rez 0 z 1 z Mellin transforms t tz 1 z Ζ z, 3 z ; Rez 1 Representations through equivalent functions With related functions With Floor For real arguments

9 9 x x x 1 ; x x x 1 x 1 ; x x 1 Χ x 1 ; x For complex arguments z Imz 1 z Χ Χ Rez 1 With Ceiling For real arguments x x x 1 ; x x x 1 x 1 ; x x 1 Χ x 1 ; x For complex arguments Rez 1 z z Χ Χ Imz 1 With IntegerPart For real arguments x int x 1 ; x x x int x 1 ; x x 1 x 1 x 1

10 x int x 1 1 Χ x 1 sgn Χ x 1 Θ x 1 ; x For complex arguments Rez 1 z int z 1 Χ Χ Imz 1 sgn Χ Rez 1 Θ Rez 1 sgn Χ Imz 1 Θ Imz 1 With FractionalPart For real arguments x x frac x 1 1 ; x x x x frac x 1 1 ; x x 1 x 1 x x x frac x 1 1 Χ x 1 sgn Χ x 1 Θ x 1 ; x For complex arguments z z frac z Χ Rez 1 Χ Imz 1 sgn Χ Rez 1 Θ Rez 1 sgn Χ Imz 1 Θ Imz 1 With Mod For real arguments x x frac x 1 1 x 1 ; x x x frac x 1 1 ; x x x x 1 mod 1 1 Χ x 1 ; x

11 11 For complex arguments Imz 1 z z Χ Χ Rez 1 1 z mod 1 With Quotient For real arguments x quotient x 1 x 1, 1 ; x x quotient x 1 x 1, 1 1 ; x quotient x 1, 1 Χ x 1 ; x For complex arguments z quotient z, 1 Χ Rez 1 Χ Imz 1 With elementary functions z z tan1 tanπ z ; z z 1 Π Zeros z 0 ; 1 Rez 1 1 Imz 1 History C. F. Gauss (1808) J. Liouville (1838) J. Hastad (1988) suggested the notation z

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