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- $$\mathrm{c}_{pq}^2(t,s;\mathbb{T})+\mathrm{s}_{pq}^2(t,s;\mathbb{T})=1,$$371 bytes (54 words) - 00:41, 15 September 2016
- [http://web.mst.edu/~bohner/sample.pdf Chapters 1-3 hosted by Martin Bohner]<br /> :Chapter 1. The Time Scales Calculus4 KB (374 words) - 15:27, 15 January 2023
- ...ev=Delta derivative of sum|next=Delta derivative of product (1)}}: Theorem 1.20 (ii)524 bytes (70 words) - 05:45, 10 June 2016
- ...a derivative of product (1)|next=Delta derivative of reciprocal}}: Theorem 1.20 (iii)572 bytes (79 words) - 05:44, 10 June 2016
- ...vative|delta differentiable]], and $f(t)f(\sigma(t)) \neq 0$. Then $\dfrac{1}{f}$ is delta differentiable and $$\left( \dfrac{1}{f} \right)^{\Delta}(t) = -\dfrac{f^{\Delta}(t)}{f(t)f(\sigma(t))},$$623 bytes (88 words) - 15:19, 21 January 2023
- ...Delta}(t)=\displaystyle\sum_{j=0}^{m-1} (\sigma(t)-\alpha)^j (t-\alpha)^{m-1-j},$$ ...nt|next=Delta derivative of reciprocal of classical polynomial}}: Theorem $1.24(i)$600 bytes (88 words) - 15:06, 25 September 2016
- ...}$, and define $g\colon \mathbb{T} \rightarrow \mathbb{R}$ by $g(t)=\dfrac{1}{(t-\alpha)^m}$. Then ...splaystyle\sum_{j=0}^{m-1} \dfrac{1}{(\sigma(t)-\alpha)^{m-j}(t-\alpha)^{j+1}},$$583 bytes (84 words) - 12:48, 23 September 2016
- $$\xi_h(z)=\dfrac{1}{h} \mathrm{Log}(1+zh),$$675 bytes (98 words) - 00:52, 30 May 2017
- [[Mozyrska-Torres logarithm at 1]]<br /> [[Mozyraska-Torres logarithm is negative on (0,1)]]<br />714 bytes (81 words) - 15:32, 21 October 2017
- ...ime scale including $1$ and at least one other point $t$ such that $0< t < 1$. The following formula holds: $$L_{\mathbb{T}}^{\Delta}(t) = \dfrac{1}{t},$$500 bytes (65 words) - 15:28, 21 October 2017
- ...nd at least one other point $t$ such that $0< t < 1$. Then $L_{\mathbb{T}}(1)=0$, where $L_{\mathbb{T}}$ denotes the [[Mozyrska-Torres logarithm]].480 bytes (63 words) - 15:28, 21 October 2017
- :1 Elements of the Time Scale Calculus ::1.1 Forward and Backward Jump Operators, Graininess Function5 KB (497 words) - 02:57, 20 December 2017
- Let $\mathbb{T}=\left\{0,\dfrac{1}{3},\dfrac{1}{2},\dfrac{7}{9},1,2,3,4,5,6,7 \right\}$.<br /> >>> ts=tsc.timescale([0,Fraction(1,3),Fraction(1,2),Fraction(7,9),1,2,3,4,5,6,7],'documentation example')4 KB (494 words) - 14:41, 4 December 2018
- ...garithm on time scales|2005|Martin Bohner|next=Euler-Cauchy logarithm}}: $(1)$588 bytes (76 words) - 17:02, 11 February 2017
- ...s logarithm on the reals|next=Mozyraska-Torres logarithm is negative on (0,1)}}438 bytes (59 words) - 15:13, 21 January 2023
- Let $\mathbb{T}$ be a [[time scale]]. If $t \in (0,1) \cap \mathbb{T}$, then $L_{\mathbb{T}}(t) < 0$, where $L_{\mathbb{T}}$ den ...res logarithm is increasing|next=Mozyrska-Torres logarithm is positive on (1,infinity)}}462 bytes (59 words) - 15:13, 21 January 2023
- Let $\mathbb{T}$ be a time scale. If $t \in (1,\infty) \cap \mathbb{T}$, then $L_{\mathbb{T}}(t) > 0$, where $L_{\mathbb{R ...r2 = Delfim F. M. Torres|prev=Mozyraska-Torres logarithm is negative on (0,1)|next=Mozyrska-Torres logarithm composed with forward jump}}465 bytes (61 words) - 15:13, 21 January 2023
- #REDIRECT [[Mozyrska-Torres logarithm is positive on (1,infinity)]]67 bytes (7 words) - 15:21, 21 October 2017
- ...thor2 = Delfim F. M. Torres|prev=Mozyrska-Torres logarithm is positive on (1,infinity)|next=Euler-Cauchy logarithm}}541 bytes (71 words) - 15:13, 21 January 2023
- :1. Introduction ::Lemma 2.1710 bytes (78 words) - 20:19, 22 January 2023
- :1 Preliminaries ::1.1 Delta Calculus1 KB (129 words) - 02:37, 20 December 2017
- :1. Introduction650 bytes (71 words) - 14:57, 21 October 2017
- ...a Mozyrska|author2 = Delfim F. M. Torres|prev=Mozyrska-Torres logarithm at 1|next=Mozyrska-Torres logarithm is increasing}}534 bytes (68 words) - 15:14, 21 January 2023
- :Chapter 1 ::1.0 Introduction1 KB (134 words) - 02:43, 20 December 2017
- ...) + \displaystyle\sum_{k=1}^{n-1} \mu(t_k) \displaystyle\prod_{m=0}^{k-1} (1+z\mu(t_m)).$$366 bytes (56 words) - 15:28, 21 January 2023
- $$\mathrm{DFT}\{f\}(z_k)=\displaystyle\sum_{k=0}^{N-1} x(t_k) \overline{e_{z_n}(t_k)} \mu(t_k),$$522 bytes (90 words) - 04:00, 26 February 2018
- :1. Introduction and motivation ::[[Partial Delta Derivative|Definition 1]]876 bytes (94 words) - 15:00, 15 January 2023
- ...\{f\}(z;s) = \displaystyle\sum_{k=-\infty}^{\infty} \dfrac{f(k)}{(1+iz)^{k+1-s}}$ ...\}(z;s) = h\displaystyle\sum_{k=-\infty}^{\infty} \dfrac{f(hk)}{(1+ihz)^{k+1-\frac{s}{h}}}$2 KB (244 words) - 20:18, 22 January 2023
- :1. Introduction550 bytes (58 words) - 02:33, 16 January 2023
- ...delta derivative|delta differentiable]] and for all $\ell \in \{0,\ldots,k-1\}$, $\displaystyle\lim_{t \rightarrow \pm \infty} f^{\Delta^{\ell}}(t)e_{\o633 bytes (89 words) - 16:50, 15 January 2023
- :1. Introduction477 bytes (60 words) - 17:47, 15 January 2023
- $$1+\mu(t)p(t) > 0.$$287 bytes (44 words) - 12:57, 16 January 2023