BetaRegularized. Notations. Primary definition. Traditional name. Traditional notation. Mathematica StandardForm notation.
|
|
|
- Βερενίκη Παπαδάκης
- 6 χρόνια πριν
- Προβολές:
Transcript
1 BetaRegularized Notations Traditional name Regularized incomplete beta function Traditional notation I z a, b Mathematica StandardForm notation BetaRegularizedz, a, b Primary definition Basic definition I z a, b za, b a, b Above generic formula cannot directly be used for nonpositive integers a, b, which lead to indeterminate expressions like 0/0. In these cases, it is more necessary to use an equivalent complete definition, presented below. Complete definition a b I z a, b z a 2F a, b; a ; z ; a b The function I z a, b can be equivalently defined through the following generalized hypergeometric function. For negative integers Α n and positive integers b m ; m n, the function I z a, b cannot be uniquely defined by a limiting procedure based on the above definition because the two variables a, b can approach integers n, m ; m n at different speeds. In the case a n, b m ; m n one defines: I z n, m m n z n m n m m m k z k k n k ; n m m n For a n and arbitrary b (except case of positive integer b m ; m n), the function I z a, b is defined as having the value :
2 I z n, b ; n b b n Specific values Specialized values For fixed z, a n a I z a, n z a k z k k ; n I z a, n 0 ; n For fixed z, b I z n, b ; n b b n I z n, b b n z n b n b b b k z k k n k ; n b b n n b I z n, b z b k z k k ; n For fixed a, b I 0 a, b 0 ; Rea I 0 a, b ; Rea I a, b ; Reb 0 General characteristics Domain and analyticity I z a, b is an analytical function of z, a, and b which is defined in z a bi z a, b Symmetries and periodicities Mirror symmetry
3 I z a, b I z a, b ; z, 0 z, Periodicity No periodicity Poles and essential singularities With respect to b For fixed z, a, the function I z a, b has only one singular point at b. It is an essential singular point ing b I z a, b, With respect to a For fixed z, b, the function I z a, b has only one singular point at a. It is an essential singular point ing a I z a, b, With respect to z For fixed a, b, the function I z a, b does not have poles and essential singularities ing z I z a, b Branch points With respect to b For fixed z, a, the function I z a, b does not have branch points b I z a, b With respect to a For fixed z, b, the function I z a, b does not have branch points a I z a, b With respect to z The function I z a, b has three branch points: z 0, z, and z z I z a, b 0,, z I z a, b, 0 log ; a a
4 z I z a, b, 0 q ; a p q p q gcdp, q z I z a, b, log ; b b z I z a, b, q ; b p q p q gcdp, q z I z a, b, log ; a b a b z I z a, b, s ; a b r s r s gcdr, s Branch cuts With respect to b For fixed z, a, the function I z a, b does not have branch cuts b I z a, b With respect to a For fixed z, b, the function I z a, b does not have branch cuts a I z a, b With respect to z For fixed a, b, the function I z a, b is a single-valued function on the z-plane cut along the intervals, 0 and,. The function I z a, b is continuous from above on the interval, 0 and from below on the interval, z I z a, b, 0,,,, lim I x Ε a, b I x a, b ; x 0 Ε lim I x Ε a, b 2 a I x a, b ; x 0 Ε lim I x Ε a, b I x a, b ; x Ε lim I x Ε a, b 2 b I x a, b 2 b sinb ; x Ε0 Series representations
5 5 Generalized power series Expansions at generic point z z 0 For the function itself I z a, b sin b a z 0 a argzz 0 z 0 a argzz 0 2 b argz 0 z arg z 0 argz 0 z a z a 0 G 2,2 a, b 2,2 z 0 z 0 b argz 0 z z 0 b argz 0 z z 0 b argz 0 z z 0 b argz 0 z z a 0 a G 2,2 a, b 2,2 z 0 G 2,2 a, b 2,2 z 0 z 0 2 b argz 0 z arg z 0 argz 0 z a a a z 0 a z z 0 2 z 0 b argz 0 z z 0 b argz 0 z z a 0 a a G 2,2 a, b 2,2 z 0 z 0 2 a G 2,2 a, b 2,2 z 0 G 2,2 a, b 2 2,2 z 0 2 z 0 2 b argz 0 z arg z 0 argz 0 z a a a 2 a a a 2 z 0 a2 z z 0 2 ; z z 0
6 I z a, b sin b a z 0 a argzz 0 z 0 a argzz 0 2 b argz 0 z arg z 0 argz 0 z a z a 0 G 2,2 a, b 2,2 z 0 z 0 b argz 0 z z 0 b argz 0 z z 0 b argz 0 z z 0 b argz 0 z z a 0 a G 2,2 a, b 2,2 z 0 G 2,2 a, b 2,2 z 0 z 0 2 b argz 0 z arg z 0 argz 0 z a a a z 0 a z z 0 2 z 0 b argz 0 z z 0 b argz 0 z z a 0 a a G 2,2 a, b 2,2 z 0 z 0 2 a G 2,2 a, b 2,2 z 0 G 2,2 a, b 2 2,2 z 0 2 z 0 2 b argz 0 z arg z 0 argz 0 z a a a 2 a a a 2 z 0 a2 z z 0 2 Oz z I z a, b sin b a z 0 a argzz 0 z 0 a argzz 0 k kj jk a kj z 0 j k j j 0 z 0 b argz 0 z z 0 b argz 0 z G 2,2 a j, b j 2,2 z 0 j argz 0 z b argz 0 z arg z 0 j a j z 0 aj z z 0 k
7 sin b I z a, b a z 0 a argzz 0 z 0 a argzz 0 k k jk a kj z 0 j k j j 0 a j z 0 aj cscb z 0 b argz 0 z z 0 b argz 0 z 2 b argz 0 z arg z 0 argz 0 z cscb a b z 0 b argz 0 z z 0 argz 0 z bbj 2 F, a b; b j ; z 0 z z 0 k ; b I z a, b sin b a z 0 a argzz 0 z 0 a argzz 0 2 b argz 0 z arg z 0 argz 0 z a z 0 a G 2,2 a, b 2,2 z 0 z 0 b argz 0 z z 0 b argz 0 z Oz z 0 Expansions on branch cuts For the function itself In the left half-plane I z a, b sin b a x a argzx argzx a x G 2,2 2 x 2 2,2 x a, b 2 2 ; z x x x 0 G 2,2 2,2 x a, b G 2,2 2,2 x x 2 a a G 2,2 2,2 x a, b a, b x G 2,2 2,2 x 2 x G 2,2 2,2 x a, b a, b z x z x I z a, b sin b a x a argzx argzx a x G 2,2 2 x 2 2,2 x a, b 2 2 Oz x 3 ; x x 0 G 2,2 2,2 x a, b G 2,2 2,2 x x 2 a a G 2,2 2,2 x a, b a, b x G 2,2 2,2 x 2 x G 2,2 2,2 x a, b a, b z x z x 2
8 8 I z a, b sin b a x a argzx argzx a x k j 0 kj a kj x jk j k j G 2,2 2,2 x a j, b j j z x k ; x x I z a, b a x a argzx argzx a x k k a kj x jk a j x aj a b x bj 2F, a b; b j ; x z x k ; x x 0 j k j j 0 I z a, b sin b a x a argzx x a argzx 2,2 G2,2 x a, b Oz x ; x x 0 In the right half-plane I z a, b sin b a 2 b argxz argx z a x b argxz x argxz b 2,2 G2,2 x a, b x a 2 x x b argxz x b argxz x argxz b argxz b x a a G 2,2 2,2 x x a2 G 2,2 2,2 x a, b a, b 2 2 x G 2,2 2,2 x x 2 a, b z x a a G 2,2 2,2 x a, b 2 x G 2,2 2,2 x a, b z x 2 ; z x x x I z a, b sin b a 2 b argxz argx z a x b argxz x argxz b 2,2 G2,2 x a, b x a 2 x x b argxz x b argxz x argxz b argxz b x a a G 2,2 2,2 x x a2 G 2,2 2,2 x a, b a, b 2 2 x G 2,2 2,2 x x 2 a, b z x a a G 2,2 2,2 x a, b 2 x G 2,2 2,2 x a, b z x 2 Oz x 3 ; x x
9 sin b k kj a kj x jk I z a, b a x a j 0 j k j x b argxz x argxz b 2,2 G2,2 x a j, b j j argx z argxz b j a j x j z x k ; x x sin b I z a, b a k k a kj x jk a j x j j k j j 0 cscb x b argxz x argxz b 2 b argxz argx z cscb a b x a x b argxz x argxz bbj 2 F, a b; b j ; x z x k ; b x x I z a, b sin b a 2 b argxz argx z a x b argxz x argxz b 2,2 G2,2 x a, b x a Oz x ; x x Expansions at z 0 For the function itself General case z a I z a, b a a, b z a I z a, b a a, b a b z a b 2 b z2 ; z 0 a a 2 a 2 a b z a b 2 b z2 Oz 3 ; a a 2 a I z a, b za a, b b k z k a k k ; z a z a I z a, b a a, b 2 F a, b; a ; z I z a, b za a b 2F a, b; a ; z b
10 z a I z a, b Oz ; a a a, b I z a, b F z, a, b ; F n z za n b k z k a, b a k k I z an b n za, b a, ba n n 3 F 2, a n, b n 2; n 2, a n 2; z n Summed form of the truncated series expansion. Generic formulas for main term a b a b 0 I z a, b z a ; z 0 True a a,b Expansions at z For the function itself General case I z a, b zb z a b a, b I z a, b zb z a b a, b a b z b a b z b a b a b z 2 ; z b b 2 b a b a b z 2 Oz 3 ; b b 2 b I z a, b zb z a k a b k z k b a, b b k ; z b I z a, b zb z a b a, b 2F, a b; b ; z a b I z a, b z b z a 2F, a b; b ; z a zb a b z I z a, b b a, b b a a 2 b z 2 ; z b 2 2 b
11 zb a b z I z a, b b a, b b a a 2 b z 2 Oz 3 ; b 2 2 b a sin a zb jk j a a b kz jk I z a, b j b k j 0 ; z b zb I z a, b b a, b F 0 2 ; a;, a b; 0 0 z, z ; b ; ; b ; zb I z a, b Oz ; b b a, b Generic formulas for main term b I z a, b ; z zb True b a,b I z a, b General case b Reb 0 zb b a,b zb b a,b Expansions at z Reb 0 ; z True z b z a b a b I z a, b csc a b sinb z a z a a b a, b 2 a b z z a b z b z a b a b I z a, b csc a b sinb z a z a a b a, b 2 a b z a b b 2 b a b 2 3 a b z 2 ; b 2 b a b 2 3 a b z 2 O z 3 ; z a z b b I z a, b csc a b sinb z a z a k a b k z k a b a, b 2 a b k k ; z a b z a z b I z a, b csc a b sinb z a z a a b a, b 2F b, a b; 2 a b; z ; z 0, a b
12 z a z b I z a, b csc a b sinb z a z a a b a, b O z ; a b I z a, b a csca b sinb b z ab ab a,b a csca b sinb b z ab ab a,b argz 0 ; a b z True I z a, b F z, a, b ; z a z b n b k a b k z k F n z, a, b a b a, b csc a b sinb z a z a I z a, b 2 a b k k b n z b z an n 2 a b n a, b 3F 2, b n 2, a b n 2; n 2, a b n 3; z n a b Summed form of the truncated series expansion. Logarithmic cases sina sina I z a, a z a z a logz Ψa a z a z a a 4 z a 2 a ; z 8 z I z a, 2 a sina za z a a a sina z a z a a 6 z a 2 a 36 z 2 I z a, a n a sina za z a a n 2 2 n z sina z a z a logz Ψa ; z a 2 a n 3 n z sina z a z a z a z na n a n k z k logz Ψn Ψa a, a n ; z n n k k I z a, a sina za z a logz Ψa a sina za z a 3F 2,, a ; 2, 2; z ; z 0, I z a, a n a sina za z a a k z k n n 2 k k sina z a z na n z a z a logz Ψn Ψa a, a n a n k z k n k k ; n z
13 a sina I z a, a n n za z a 3F 2,, a ; 2, n 2; z sina z a z na n z a z a logz Ψn Ψa a, a n a n k z k n k k ; n z 0, sina z a z b I z a, b z a z a logz Ψa Ψa b a b a, b O z ; a b sin a a sin a I z a, b z a z a logz Ψa z a z a O z ; a b I z a, b b z ab a logz sina ab a,b b z ab a logz sina ab a,b argz 0 ; a b z True I z a, b a logz sina a logz sina argz 0 ; a b z True Generic formulas for main term I z a, b z a z b sina za z a logzψaψab ab a,b sin a z a z a logzψa a sin a za z a z a z b ab a,b csc a b sinb za z a a b a b ; z True I z a, b a ab z ab ab a,b a sina a zz logzψa z sina logzψaψab a ab z ab csca b sinb argz 0 ab a,b b z ab sina a logzψaψab ab a,b a sina a zz logzψa z b z ab ab a,b a csca b sinb argz 0 a b argz 0 a b ; z argz 0 a b argz 0 a b True Expansions at generic point a a 0 For the function itself
14 I z a, b I z a 0, b a 2 0 z a 0, b logz Ψb a 0 Ψa 0 z a 0 3 F 2 a 0, a 0, b; a 0, a 0 ; z a a 0 a 2 0 a 0, b 2 a 3 0 a 0, b 2 4 F 3 b, a 0, a 0, a 0 ; a 0, a 0, a 0 ; z 3 F 2 b, a 0, a 0 ; a 0, a 0 ; z logz Ψa 0 Ψb a 0 a I z a, b I z a 0, b z a 0 z a 0, b log 2 z Ψa 0 Ψb a 0 2 logz Ψa 0 Ψb a 0 Ψ a 0 Ψ b a 0 a 0 3 a a 0 2 ; a a 0 a 2 0 z a 0, b logz Ψb a 0 Ψa 0 z a 0 3 F 2 a 0, a 0, b; a 0, a 0 ; z a a 0 a 2 0 a 0, b 2 a 3 0 a 0, b 2 4 F 3 b, a 0, a 0, a 0 ; a 0, a 0, a 0 ; z 3 F 2 b, a 0, a 0 ; a 0, a 0 ; z logz Ψa 0 Ψb a 0 a 0 z a 0 z a 0, b log 2 z Ψa 0 Ψb a 0 2 logz Ψa 0 Ψb a 0 Ψ a 0 Ψ b a 0 a 0 3 a a 0 2 Oa a I z a, b b sin b a 0 z a 0 k kj log j z b a 0 jk kj2 F kj b, d, d 2,, d kj ; d, d 2,, d kj ; j 0 j j i i2f i b, c, c 2,, c i ; c, c 2,, c i ; z a 0 logz i 0 d d 2 d k b a 0 c c 2 c k a 0 k i a a 0 k ; I z a, b I z a 0, b Oa a 0 Expansions at generic point b b 0 For the function itself I z a, b I z a, b 0 zb 0 a b 0 a b 0 3 F 2 a, b 0, b 0 ; b 0, b 0 ; z log z Ψb 0 Ψa b 0 2 F a, b 0 ; b 0 ; z b b 0 b F 3 a, b 0, b 0, b 0 ; b 0, b 0, b 0 ; z b 0 log z Ψb 0 Ψa b 0 3 F 2 a, b 0, b 0 ; b 0, b 0 ; z 2 log2 z Ψb 0 Ψa b 0 2 log z Ψb 0 Ψa b 0 Ψ b 0 Ψ a b 0 2F a, b 0 ; b 0 ; z b b 0 2 ; b b 0
15 I z a, b I z a, b 0 zb 0 a b 0 a b 0 3 F 2 a, b 0, b 0 ; b 0, b 0 ; z log z Ψb 0 Ψa b 0 2 F a, b 0 ; b 0 ; z b b 0 b F 3 a, b 0, b 0, b 0 ; b 0, b 0, b 0 ; z b 0 log z Ψb 0 Ψa b 0 3 F 2 a, b 0, b 0 ; b 0, b 0 ; z 2 log2 z Ψb 0 Ψa b 0 2 log z Ψb 0 Ψa b 0 Ψ b 0 Ψ a b 0 2F a, b 0 ; b 0 ; z b b 0 2 Ob b I z a, b I z a, b 0 a sin a b 0 z b 0 k kj log j z a b 0 kj kj2f kja, d, d 2,, d kj ; d, d 2,, d kj ; k j 0 j j i i2f i a, c, c 2,, c i ; c, c 2,, c i ; z b 0 log z i 0 d d 2 d k a b 0 c c 2 c k b 0 k i b b 0 k ; I z a, b I z a, b 0 Ob b 0 Residue representations a b sin b za I z a, b res s a j 0 b s z s s j ; z a s a b sin b za I z a, b a z a b j 0 b s res s s z s a s a j 0 b s res s s z s a s j b ; I z a, a n n n sina z a a a n s s res s s z a s a n res s j 0 s z s a s a n s a j n res s j n s z s a n s a j n ; z n a s Integral representations On the real axis
16 6 Of the direct function I z a, b z a, b t a t b t ; Rea I z a, b z Rea a, b b t a t b k z k 0 k Rea b k z ak t a k k Contour integral representations a b sin b za s a s b s I z a, b z s s ; argz 2 2 a a s sin b za I z a, b s b s a s b s z s s ; arg z 2 a a b sin b za Γ s a s b s I z a, b z s s ; 0 Γ minrea, Reb argz 2 2 a Γ a s sin b za Γ I z a, b s b s a s b s z s s ; 2 a Γ max0, Reb Γ minrea, Reb arg z Continued fraction representations I z a, b za z b a a, b r r2 r3 r4 r5 a k a b k z k b k z ; r2 k r2 k a 2 k a 2 k a 2 k a 2 k z a z b a k a b k z k b k z I z a, b ; r2 k r2 k a a, b k rk, a 2 k a 2 k a 2 k a 2 k Differential equations Ordinary linear differential equations and wronskians For the direct function itself
17 z z w z a a b 2 z w z 0 ; wz c I z a, b c W z, I z a, b zb z a a, b w z a a b 2 gz g z gz gz g z g z w z 0 ; wz c c 2 I gz a, b W z, I gz a, b gzb gz a g z a, b a a b 2 gz g z w z hz 2 h z gz gz hz g z hz 2 hz g z 2 h z 2 a a b 2 gz g z h z hz 2 gz gz hz w z g z h z hz g z h z hz wz 0 ; wz c hz c 2 hz I gz a, b W z hz, hz I gz a, b gzb gz a hz 2 g z a, b d z r z 2 w z d a b r 2 s z r a r 2 s z w z s b d r z r a r s d z r wz 0 ; wz c z s c 2 z s I d z ra, b W z z s, z s I d z ra, b r z2 s d z r a d z r b a, b d r z w z d a b logr 2 logs r z a logr 2 logs w z logs d a b logr logs r z a logr logs wz 0 ; wz c s z c 2 s z I d r za, b W z s z, s z I d r za, b d rz d r z a d r z b s 2 z logr a, b Transformations Transformations and argument simplifications Argument involving basic arithmetic operations I z a, b I z b, a
18 a b I z a, b I z a, b a b zb z a a b I z a, b I z a, b z b z a a b n I z a n, b I z a, b a b n a n, b a n k z b z akn 2 a b n k ; n n I z a n, b I z a, b a b a, b a k z b z ak 2 a b k ; n Identities Recurrence identities Consecutive neighbors a b I z a, b I z a, b a b zb z a a b I z a, b I z a, b z b z a a b Distant neighbors z a z b n a b k z k I z a, b I z a n, b a, b a b a k k ; n z a z b n I z a, b I z a n, b a, b a b a k z k ; n 2 a b k Functional identities Relations between contiguous functions I z a, b a b a I za, b b I z a, b I z a, b z I z a, b z I z a, b Additional relations between contiguous functions
19 I z a, b a b a z a z I za, b b I z a, b Major general cases I z a, b I z b, a I z a, b z a z a sina z ab z ab I a b, b sinb csca b ; a b z 0, z Differentiation Low-order differentiation With respect to z I z a, b z zb z a a, b 2 I z a, b z With respect to a zb2 za2 z a b 2 z a, b I z a, b a a b logz Ψa Ψa bi z a, b z a 3 F 2a, a, b; a, a ; z a b 2 I z a, b a za a a b b a 4 F 3a, a, a, b; a, a, a ; z logz Ψa Ψa b 3 F 2a, a, b; a, a ; z I z a, b log 2 z 2 Ψa b logz Ψa 2 Ψa b 2 2 Ψa logz Ψa b Ψ a Ψ a b With respect to b I z a, b b a b z b 3F 2b, b, a; b, b ; z I z b, a Ψb Ψa b log z b a 2 I z a, b a zb b a b a log z Ψb Ψa b 3 F 2b, b, a; b, b ; z b 4 F 3b, b, b, a; b, b, b ; z I z b, a log 2 z 2 Ψa b log z Ψb 2 Ψa b 2 2 Ψb log z Ψa b Ψ b Ψ a b Symbolic differentiation
20 20 With respect to z n I z a, b zb zan n n I z a, b nk n z n a, b k b k a nk z z k ; n n I z a, b z n n a b z bn z a 2 F a, n; b n ; a z ; n With respect to a n I z a, b a n b sin b a n za n nj log j z a b nj nj2f nja, a 2,, a nj, b; a, a 2,, a nj ; j 0 j j i i2 F ic, c 2,, c i, b; c, c 2,, c i ; z a logz i 0 a a 2 a n a b c c 2 c n a n i ; With respect to b n I z a, b a sin a b n zb n b n n nj log j z a b nj nj2f nja, a 2,, a nj, ba; a, a 2,, a nj ; j 0 j j i i2 F ic, c 2,, c i, a; c, c 2,, c i ; z b log z i 0 a a 2 a n a b c c 2 c n b n i ; Fractional integro-differentiation With respect to z Α I z a, b a b z aα 2F a, b; a Α ; z ; a z Α b Α I z a, b b k Α exp z, a kz akα z Α a, b ; z a k k Integration Indefinite integration
21 2 Involving only one direct function a I a z a, b z z I a z a, b a a b I a za, b I z a, b z z I z a, b a a b I za, b Involving one direct function and elementary functions Involving power function z Α I z a, b z zα Α I za, b z Α I z a, b z zα Α I za, b a b a Α Α a a b Α I za Α, b a b a Α Α a a b Α I za Α, b I z a, b z a z z a 2 a, b 3 F 2 a, a, b; a, a ; z z b z a I z a, b z 2 za, b I z a, b z n I z a, b z za, b n z n z a, b n a, b Involving rational function I z a, b z zb 3F 2 b, b, a; b, b ; z z b 2 a, b I z b, a I z a, b log z Integral transforms Fourier cos transforms c t I t a, bx 2b 2 a sin b a b 2 a 3,2 G x2 2,4 4 0, ab a 2, a 2, ab 2 2, 2 ; x Rea Fourier sin transforms
22 s t I t a, bx 2b 2 a sin b a b 2 a 3,2 sgnxg x2 2,4 4 ab, 2 a 2, a 2, ab 2 2, 0 ; x Rea 2 Laplace transforms t I t a, bz a a b Ua, a b, z ; Rez 0 Rea z b Mellin transforms cot a z sin b a b z a b t I t a, bz ; Rea z 0 Rez 0 Rea b z z a z a Representations through more general functions Through hypergeometric functions Involving 2 F a b I z a, b z a 2F a, b; a ; z ; a b a b I z a, b z b z a 2F, a b; a ; z ; a b a b I z a, b z b z a 2 F, a b; b ; z ; b a csc a b I z a, b csc a b sinb z a z a z b z a 2F b, a b ; a b 2; a b z ; a b Involving 2 F z a I z a, b a a, b 2F a, b; a ; z ; a zb I z a, b a a, b za 2F, a b; a ; z ; a I z a, b zb z a b a, b 2F, a b; b ; z ; b
23 z a z b I z a, b a b a, b 2F b, a b; 2 a b; z csc a b sinb z a z a ; a b Through hypergeometric functions of two variables zb I z a, b b a, b F 0 2 ; a;, a b; 0 0 z, z ; b ; ; b ; Through Meijer G Classical cases for the direct function itself sin b a b za I z a, b G,2 2,2 z a a, b 0, a Classical cases involving algebraic functions za z b I z a, b b G 2,2,2 z 0, a b 0, a z b I a, b zab z b G 2, 2,2 z, a, a b ; z, 0 Classical cases involving algebraic functions in the arguments I z a, b z a b G,2 2,2 z, b a, 0 ; z, sin b a b z ab I z a, b G,2 2,2 z z a, a b a, 0 ; z, Through other functions Involving some hypergeometric-type functions I z a, b I,z a, b ; Reb I z a, b a, b,za, b ; Reb 0 Representations through equivalent functions With inverse function
24 I Iz a, b z a,b I I,z2 a,b a, b z 2 ; z 2 0 With related functions a b I z a, b a b za, b
25 25 Copyright This document was downloaded from functions.wolfram.com, a comprehensive online compendium of formulas involving the special functions of mathematics. For a key to the notations used here, see Please cite this document by referring to the functions.wolfram.com page from which it was downloaded, for example: To refer to a particular formula, cite functions.wolfram.com followed by the citation number. e.g.: This document is currently in a preliminary form. If you have comments or suggestions, please comments@functions.wolfram.com , Wolfram Research, Inc.
ExpIntegralE. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation
ExpIntegralE Notations Traditional name Exponential integral E Traditional notation E Mathematica StandardForm notation ExpIntegralE, Primary definition 06.34.0.000.0 E t t t ; Re 0 Specific values Specialied
Notations. Primary definition. Specific values. General characteristics. Traditional name. Traditional notation. Mathematica StandardForm notation
KelvinKei Notations Traditional name Kelvin function of the second kind Traditional notation kei Mathematica StandardForm notation KelvinKei Primary definition 03.5.0.000.0 kei kei 0 Specific values Values
DiracDelta. Notations. Primary definition. Specific values. General characteristics. Traditional name. Traditional notation
DiracDelta Notations Traditional name Dirac delta function Traditional notation x Mathematica StandardForm notation DiracDeltax Primary definition 4.03.02.000.0 x Π lim ε ; x ε0 x 2 2 ε Specific values
GegenbauerC3General. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation
GegenbauerC3General Notations Traditional name Gegenbauer function Traditional notation C Ν Λ z Mathematica StandardForm notation GegenbauerCΝ, Λ, z Primary definition 07.4.0.000.0 C Λ Ν z Λ Π Ν Λ F Ν,
Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation. Specialized values
PolyGamma Notations Traditional name Digamma function Traditional notation Ψz Mathematica StandardForm notation PolyGammaz Primary definition 06.4.02.000.0 Ψz k k k z Specific values Specialized values
Notations. Primary definition. Specific values. General characteristics. Series representations. Traditional name. Traditional notation
Pi Notations Traditional name Π Traditional notation Π Mathematica StandardForm notation Pi Primary definition.3... Π Specific values.3.3.. Π 3.5965358979338663383795889769399375589795937866868998683853
Factorial. Notations. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation. Specialized values
Factorial Notatios Traditioal ame Factorial Traditioal otatio Mathematica StadardForm otatio Factorial Specific values Specialized values 06.0.0.000.0 k ; k 06.0.0.000.0 ; 06.0.0.000.0 p q q p q p k q
Section 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
Inverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------
Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin
PARTIAL NOTES for 6.1 Trigonometric Identities
PARTIAL NOTES for 6.1 Trigonometric Identities tanθ = sinθ cosθ cotθ = cosθ sinθ BASIC IDENTITIES cscθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ PYTHAGOREAN IDENTITIES sin θ + cos θ =1 tan θ +1= sec θ 1 + cot
Trigonometric Formula Sheet
Trigonometric Formula Sheet Definition of the Trig Functions Right Triangle Definition Assume that: 0 < θ < or 0 < θ < 90 Unit Circle Definition Assume θ can be any angle. y x, y hypotenuse opposite θ
Homework 3 Solutions
Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For
If we restrict the domain of y = sin x to [ π, π ], the restrict function. y = sin x, π 2 x π 2
![If we restrict the domain of y = sin x to [ π, π ], the restrict function. y = sin x, π 2 x π 2](/thumbs/53/31933086.jpg "If we restrict the domain of y = sin x to [ π, π ], the restrict function. y = sin x, π 2 x π 2")
Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the
If we restrict the domain of y = sin x to [ π 2, π 2
Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the
2 Composition. Invertible Mappings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,
CHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS
CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =
HermiteHGeneral. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation
HermiteHGeeral Notatios Traditioal ame Hermite fuctio Traditioal otatio H Mathematica StadardForm otatio HermiteH, Primary defiitio 07.0.0.000.0 H F ; ; F ; 3 ; Specific values Specialied values For fixed
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all
D Alembert s Solution to the Wave Equation
D Alembert s Solution to the Wave Equation MATH 467 Partial Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Objectives In this lesson we will learn: a change of variable technique
Introductions to EllipticThetaPrime4
Introductions to EllipticThetaPrie Introduction to the Jacobi theta functions General The basic achieveents in studying infinite series were ade in the 18th and 19th centuries when atheaticians investigated
CRASH COURSE IN PRECALCULUS
CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter
Chapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Math221: HW# 1 solutions
Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin
An Inventory of Continuous Distributions
Appendi A An Inventory of Continuous Distributions A.1 Introduction The incomplete gamma function is given by Also, define Γ(α; ) = 1 with = G(α; ) = Z 0 Z 0 Z t α 1 e t dt, α > 0, >0 t α 1 e t dt, α >
A summation formula ramified with hypergeometric function and involving recurrence relation
South Asian Journal of Mathematics 017, Vol. 7 ( 1): 1 4 www.sajm-online.com ISSN 51-151 RESEARCH ARTICLE A summation formula ramified with hypergeometric function and involving recurrence relation Salahuddin
Example Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
derivation of the Laplacian from rectangular to spherical coordinates
derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used
Fourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)
Practice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1
Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the
Reminders: linear functions
Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U
Approximation of distance between locations on earth given by latitude and longitude
Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth
Partial Differential Equations in Biology The boundary element method. March 26, 2013
The boundary element method March 26, 203 Introduction and notation The problem: u = f in D R d u = ϕ in Γ D u n = g on Γ N, where D = Γ D Γ N, Γ D Γ N = (possibly, Γ D = [Neumann problem] or Γ N = [Dirichlet
SPECIAL FUNCTIONS and POLYNOMIALS
SPECIAL FUNCTIONS and POLYNOMIALS Gerard t Hooft Stefan Nobbenhuis Institute for Theoretical Physics Utrecht University, Leuvenlaan 4 3584 CC Utrecht, the Netherlands and Spinoza Institute Postbox 8.195
k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +
[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +](/thumbs/73/69566903.jpg "k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +")
Chapter 3. Fuzzy Arithmetic 3- Fuzzy arithmetic: ~Addition(+) and subtraction (-): Let A = [a and B = [b, b in R If x [a and y [b, b than x+y [a +b +b Symbolically,we write A(+)B = [a (+)[b, b = [a +b
Section 9.2 Polar Equations and Graphs
180 Section 9. Polar Equations and Graphs In this section, we will be graphing polar equations on a polar grid. In the first few examples, we will write the polar equation in rectangular form to help identify
9.09. # 1. Area inside the oval limaçon r = cos θ. To graph, start with θ = 0 so r = 6. Compute dr
9.9 #. Area inside the oval limaçon r = + cos. To graph, start with = so r =. Compute d = sin. Interesting points are where d vanishes, or at =,,, etc. For these values of we compute r:,,, and the values
Homework 8 Model Solution Section
MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx
ANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?
Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least
Concrete Mathematics Exercises from 30 September 2016
Concrete Mathematics Exercises from 30 September 2016 Silvio Capobianco Exercise 1.7 Let H(n) = J(n + 1) J(n). Equation (1.8) tells us that H(2n) = 2, and H(2n+1) = J(2n+2) J(2n+1) = (2J(n+1) 1) (2J(n)+1)
3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β
3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle
Matrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Mock Exam 7. 1 Hong Kong Educational Publishing Company. Section A 1. Reference: HKDSE Math M Q2 (a) (1 + kx) n 1M + 1A = (1) =
Mock Eam 7 Mock Eam 7 Section A. Reference: HKDSE Math M 0 Q (a) ( + k) n nn ( )( k) + nk ( ) + + nn ( ) k + nk + + + A nk... () nn ( ) k... () From (), k...() n Substituting () into (), nn ( ) n 76n 76n
Aquinas College. Edexcel Mathematical formulae and statistics tables DO NOT WRITE ON THIS BOOKLET
Aquinas College Edexcel Mathematical formulae and statistics tables DO NOT WRITE ON THIS BOOKLET Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE in Mathematics and Further Mathematics Mathematical
Areas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
The Simply Typed Lambda Calculus
Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and
Second Order Partial Differential Equations
Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y
C.S. 430 Assignment 6, Sample Solutions
C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order
Chapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
6.1. Dirac Equation. Hamiltonian. Dirac Eq.
6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2
MathCity.org Merging man and maths
MathCity.org Merging man and maths Exercise 10. (s) Page Textbook of Algebra and Trigonometry for Class XI Available online @, Version:.0 Question # 1 Find the values of sin, and tan when: 1 π (i) (ii)
SCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018
Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 SCITECH Volume 3, Issue 2 RESEARCH ORGANISATION ublished online: March 29, 208 Journal of rogressive Research in Mathematics www.scitecresearch.com/journals
Solutions to Exercise Sheet 5
Solutions to Eercise Sheet 5 jacques@ucsd.edu. Let X and Y be random variables with joint pdf f(, y) = 3y( + y) where and y. Determine each of the following probabilities. Solutions. a. P (X ). b. P (X
Areas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Arithmetical applications of lagrangian interpolation. Tanguy Rivoal. Institut Fourier CNRS and Université de Grenoble 1
Arithmetical applications of lagrangian interpolation Tanguy Rivoal Institut Fourier CNRS and Université de Grenoble Conference Diophantine and Analytic Problems in Number Theory, The 00th anniversary
Exercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.
Exercises 0 More exercises are available in Elementary Differential Equations. If you have a problem to solve any of them, feel free to come to office hour. Problem Find a fundamental matrix of the given
4.6 Autoregressive Moving Average Model ARMA(1,1)
84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this
Commutative Monoids in Intuitionistic Fuzzy Sets
Commutative Monoids in Intuitionistic Fuzzy Sets S K Mala #1, Dr. MM Shanmugapriya *2 1 PhD Scholar in Mathematics, Karpagam University, Coimbatore, Tamilnadu- 641021 Assistant Professor of Mathematics,
Chapter 6 BLM Answers
Chapter 6 BLM Answers BLM 6 Chapter 6 Prerequisite Skills. a) i) II ii) IV iii) III i) 5 ii) 7 iii) 7. a) 0, c) 88.,.6, 59.6 d). a) 5 + 60 n; 7 + n, c). rad + n rad; 7 9,. a) 5 6 c) 69. d) 0.88 5. a) negative
Other Test Constructions: Likelihood Ratio & Bayes Tests
Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :
2. THEORY OF EQUATIONS. PREVIOUS EAMCET Bits.
EAMCET-. THEORY OF EQUATIONS PREVIOUS EAMCET Bits. Each of the roots of the equation x 6x + 6x 5= are increased by k so that the new transformed equation does not contain term. Then k =... - 4. - Sol.
Trigonometry Functions (5B) Young Won Lim 7/24/14
Trigonometry Functions (5B 7/4/14 Copyright (c 011-014 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
Laplace Expansion. Peter McCullagh. WHOA-PSI, St Louis August, Department of Statistics University of Chicago
Laplace Expansion Peter McCullagh Department of Statistics University of Chicago WHOA-PSI, St Louis August, 2017 Outline Laplace approximation in 1D Laplace expansion in 1D Laplace expansion in R p Formal
Pg The perimeter is P = 3x The area of a triangle is. where b is the base, h is the height. In our case b = x, then the area is
Pg. 9. The perimeter is P = The area of a triangle is A = bh where b is the base, h is the height 0 h= btan 60 = b = b In our case b =, then the area is A = = 0. By Pythagorean theorem a + a = d a a =
Fundamentals of Signals, Systems and Filtering
Fundamentals of Signals, Systems and Filtering Brett Ninness c 2000-2005, Brett Ninness, School of Electrical Engineering and Computer Science The University of Newcastle, Australia. 2 c Brett Ninness
Written Examination. Antennas and Propagation (AA ) April 26, 2017.
Written Examination Antennas and Propagation (AA. 6-7) April 6, 7. Problem ( points) Let us consider a wire antenna as in Fig. characterized by a z-oriented linear filamentary current I(z) = I cos(kz)ẑ
Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)
Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts
DESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0.
DESIGN OF MACHINERY SOLUTION MANUAL -7-1! PROBLEM -7 Statement: Design a double-dwell cam to move a follower from to 25 6, dwell for 12, fall 25 and dwell for the remader The total cycle must take 4 sec
SCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions
SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)
ω ω ω ω ω ω+2 ω ω+2 + ω ω ω ω+2 + ω ω+1 ω ω+2 2 ω ω ω ω ω ω ω ω+1 ω ω2 ω ω2 + ω ω ω2 + ω ω ω ω2 + ω ω+1 ω ω2 + ω ω+1 + ω ω ω ω2 + ω
0 1 2 3 4 5 6 ω ω + 1 ω + 2 ω + 3 ω + 4 ω2 ω2 + 1 ω2 + 2 ω2 + 3 ω3 ω3 + 1 ω3 + 2 ω4 ω4 + 1 ω5 ω 2 ω 2 + 1 ω 2 + 2 ω 2 + ω ω 2 + ω + 1 ω 2 + ω2 ω 2 2 ω 2 2 + 1 ω 2 2 + ω ω 2 3 ω 3 ω 3 + 1 ω 3 + ω ω 3 +
MATHEMATICS. 1. If A and B are square matrices of order 3 such that A = -1, B =3, then 3AB = 1) -9 2) -27 3) -81 4) 81
1. If A and B are square matrices of order 3 such that A = -1, B =3, then 3AB = 1) -9 2) -27 3) -81 4) 81 We know that KA = A If A is n th Order 3AB =3 3 A. B = 27 1 3 = 81 3 2. If A= 2 1 0 0 2 1 then
Partial Trace and Partial Transpose
Partial Trace and Partial Transpose by José Luis Gómez-Muñoz http://homepage.cem.itesm.mx/lgomez/quantum/ jose.luis.gomez@itesm.mx This document is based on suggestions by Anirban Das Introduction This
TMA4115 Matematikk 3
TMA4115 Matematikk 3 Andrew Stacey Norges Teknisk-Naturvitenskapelige Universitet Trondheim Spring 2010 Lecture 12: Mathematics Marvellous Matrices Andrew Stacey Norges Teknisk-Naturvitenskapelige Universitet
ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Αν κάπου κάνετε κάποιες υποθέσεις να αναφερθούν στη σχετική ερώτηση. Όλα τα αρχεία που αναφέρονται στα προβλήματα βρίσκονται στον ίδιο φάκελο με το εκτελέσιμο
Instruction Execution Times
1 C Execution Times InThisAppendix... Introduction DL330 Execution Times DL330P Execution Times DL340 Execution Times C-2 Execution Times Introduction Data Registers This appendix contains several tables
Section 7.6 Double and Half Angle Formulas
09 Section 7. Double and Half Angle Fmulas To derive the double-angles fmulas, we will use the sum of two angles fmulas that we developed in the last section. We will let α θ and β θ: cos(θ) cos(θ + θ)
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal
Συντακτικές λειτουργίες
2 Συντακτικές λειτουργίες (Syntactic functions) A. Πτώσεις και συντακτικές λειτουργίες (Cases and syntactic functions) The subject can be identified by asking ποιος (who) or τι (what) the sentence is about.
Differential equations
Differential equations Differential equations: An equation inoling one dependent ariable and its deriaties w. r. t one or more independent ariables is called a differential equation. Order of differential
Strain gauge and rosettes
Strain gauge and rosettes Introduction A strain gauge is a device which is used to measure strain (deformation) on an object subjected to forces. Strain can be measured using various types of devices classified
Uniform Convergence of Fourier Series Michael Taylor
Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula
Review Test 3. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Review Test MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. 1) sin - 11π 1 1) + - + - - ) sin 11π 1 ) ( -
Lecture 26: Circular domains
Introductory lecture notes on Partial Differential Equations - c Anthony Peirce. Not to be copied, used, or revised without eplicit written permission from the copyright owner. 1 Lecture 6: Circular domains
CHAPTER 101 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD
CHAPTER FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD EXERCISE 36 Page 66. Determine the Fourier series for the periodic function: f(x), when x +, when x which is periodic outside this rge of period.
Trigonometry 1.TRIGONOMETRIC RATIOS
Trigonometry.TRIGONOMETRIC RATIOS. If a ray OP makes an angle with the positive direction of X-axis then y x i) Sin ii) cos r r iii) tan x y (x 0) iv) cot y x (y 0) y P v) sec x r (x 0) vi) cosec y r (y
A Two-Sided Laplace Inversion Algorithm with Computable Error Bounds and Its Applications in Financial Engineering
Electronic Companion A Two-Sie Laplace Inversion Algorithm with Computable Error Bouns an Its Applications in Financial Engineering Ning Cai, S. G. Kou, Zongjian Liu HKUST an Columbia University Appenix
Εργαστήριο Ανάπτυξης Εφαρμογών Βάσεων Δεδομένων. Εξάμηνο 7 ο
Εργαστήριο Ανάπτυξης Εφαρμογών Βάσεων Δεδομένων Εξάμηνο 7 ο Procedures and Functions Stored procedures and functions are named blocks of code that enable you to group and organize a series of SQL and PL/SQL
Potential Dividers. 46 minutes. 46 marks. Page 1 of 11
Potential Dividers 46 minutes 46 marks Page 1 of 11 Q1. In the circuit shown in the figure below, the battery, of negligible internal resistance, has an emf of 30 V. The pd across the lamp is 6.0 V and
Distances in Sierpiński Triangle Graphs
Distances in Sierpiński Triangle Graphs Sara Sabrina Zemljič joint work with Andreas M. Hinz June 18th 2015 Motivation Sierpiński triangle introduced by Wac law Sierpiński in 1915. S. S. Zemljič 1 Motivation
Q1a. HeavisideTheta x. Plot f, x, Pi, Pi. Simplify, n Integers
2 M2 Fourier Series answers in Mathematica Note the function HeavisideTheta is for x>0 and 0 for x
Every set of first-order formulas is equivalent to an independent set
Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent
g-selberg integrals MV Conjecture An A 2 Selberg integral Summary Long Live the King Ole Warnaar Department of Mathematics Long Live the King
Ole Warnaar Department of Mathematics g-selberg integrals The Selberg integral corresponds to the following k-dimensional generalisation of the beta integral: D Here and k t α 1 i (1 t i ) β 1 1 i
Main source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1
Main source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1 A Brief History of Sampling Research 1915 - Edmund Taylor Whittaker (1873-1956) devised a
Bayesian statistics. DS GA 1002 Probability and Statistics for Data Science.
Bayesian statistics DS GA 1002 Probability and Statistics for Data Science http://www.cims.nyu.edu/~cfgranda/pages/dsga1002_fall17 Carlos Fernandez-Granda Frequentist vs Bayesian statistics In frequentist
ECE Spring Prof. David R. Jackson ECE Dept. Notes 2
ECE 634 Spring 6 Prof. David R. Jackson ECE Dept. Notes Fields in a Source-Free Region Example: Radiation from an aperture y PEC E t x Aperture Assume the following choice of vector potentials: A F = =
10/3/ revolution = 360 = 2 π radians = = x. 2π = x = 360 = : Measures of Angles and Rotations
//.: Measures of Angles and Rotations I. Vocabulary A A. Angle the union of two rays with a common endpoint B. BA and BC C. B is the vertex. B C D. You can think of BA as the rotation of (clockwise) with
Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in
Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in : tail in X, head in A nowhere-zero Γ-flow is a Γ-circulation such that
Απόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.
Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο The time integral of a force is referred to as impulse, is determined by and is obtained from: Newton s 2 nd Law of motion states that the action
Answers - Worksheet A ALGEBRA PMT. 1 a = 7 b = 11 c = 1 3. e = 0.1 f = 0.3 g = 2 h = 10 i = 3 j = d = k = 3 1. = 1 or 0.5 l =
C ALGEBRA Answers - Worksheet A a 7 b c d e 0. f 0. g h 0 i j k 6 8 or 0. l or 8 a 7 b 0 c 7 d 6 e f g 6 h 8 8 i 6 j k 6 l a 9 b c d 9 7 e 00 0 f 8 9 a b 7 7 c 6 d 9 e 6 6 f 6 8 g 9 h 0 0 i j 6 7 7 k 9
Second Order RLC Filters
ECEN 60 Circuits/Electronics Spring 007-0-07 P. Mathys Second Order RLC Filters RLC Lowpass Filter A passive RLC lowpass filter (LPF) circuit is shown in the following schematic. R L C v O (t) Using phasor
Η ΨΥΧΙΑΤΡΙΚΗ - ΨΥΧΟΛΟΓΙΚΗ ΠΡΑΓΜΑΤΟΓΝΩΜΟΣΥΝΗ ΣΤΗΝ ΠΟΙΝΙΚΗ ΔΙΚΗ
ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΝΟΜΙΚΗ ΣΧΟΛΗ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ ΤΟΜΕΑΣ ΙΣΤΟΡΙΑΣ ΦΙΛΟΣΟΦΙΑΣ ΚΑΙ ΚΟΙΝΩΝΙΟΛΟΓΙΑΣ ΤΟΥ ΔΙΚΑΙΟΥ Διπλωματική εργασία στο μάθημα «ΚΟΙΝΩΝΙΟΛΟΓΙΑ ΤΟΥ ΔΙΚΑΙΟΥ»