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• Delta simple useful formula
  $$e_p(\sigma(t),s;\mathbb{T})=(1+\mu(t)p(t))e_p(t,s;\mathbb{T}),$$ ...ev=Delta derivative at right-dense|next=Delta derivative of sum}}: Theorem 1.16 (iv)
  645 bytes (97 words) - 06:08, 10 June 2016
• Delta differentiable implies continuous
  Supposing \( f \) is differentiable, given that, by Definition 1.10, $ \epsilon^* > 0 $ there is a neighborhood \( U \) \( (U = (t - \delta, ...gative image) and considering (without loss of generality) \( \epsilon^* < 1 \), it follows that
  4 KB (666 words) - 01:14, 15 March 2022
• Derivation of delta e sub p on T=Z
  e_p(t,s) &= \exp \left( \displaystyle\int_{s}^{t} \dfrac{1}{\mu(\tau)} \log(1 + p(\tau)) \Delta \tau \right) \\ &= \exp \left( \displaystyle\sum_{k=s}^{t-1} \log(1+p(k)) \right) \\
  512 bytes (92 words) - 19:33, 29 April 2015
• Delta gk
  g_0(t,s)=1 \\ g_{k+1}(t,s)=\displaystyle\int_s^t g_k(\sigma(\tau),s) \Delta \tau.
  812 bytes (134 words) - 14:13, 28 January 2023
• Uniform distribution
  \dfrac{1}{\sigma(b)-a} &; a \leq t \leq b \\
  515 bytes (74 words) - 01:22, 30 September 2018
• Gamma distribution
  ...bb{R}$ with $\lambda > 0$ and define $\Lambda_0(t,t_0)=0, \Lambda_1(t,t_0)=1$ and $$\Lambda_{k+1}(t,t_0) = -\displaystyle\int_{t_0}^t (\ominus \lambda)(\tau) \Lambda_k(\sig
  626 bytes (90 words) - 14:11, 28 January 2023
• Probability density function
  $$\displaystyle\int_0^{\infty} f(t) \Delta t = 1.$$ ...umsystem.edu/xmlui/bitstream/handle/10355/29595/Matthews_2011.pdf?sequence=1 Probability theory on time scales and applications to finance and inequalit
  578 bytes (80 words) - 21:44, 14 April 2015
• Cylinder strip
  The cylinder strip $\mathbb{Z}_h$ is defined for $h>1$ by
  461 bytes (70 words) - 00:50, 30 May 2017
• Bilateral Laplace transform
  ...=Ian A. Gravagne|author3=Robert J. Marks II|prev=findme|next=findme}}: $(3.1)$ ...plications|2021|Tom Cuchta|author2=Svetlin Georgiev|prev=|next=}}: Section 1
  927 bytes (130 words) - 15:12, 21 January 2023
• L'Hospital's Rule
  $$L_{\epsilon}(\infty) := \left\{t \in \mathbb{T} \colon t > \dfrac{1}{\epsilon} \right\}$$ <strong>Theorem (L'Hospital's Rule 1):</strong> Assume $f,g$ $\Delta$-differentiable on $\mathbb{T}$ and for som
  2 KB (360 words) - 08:11, 8 February 2015
• Delta Taylor's formula
  ...entiable on $\mathbb{T}^{\kappa^n}$. Let $\alpha \in \mathbb{T}^{\kappa^{n-1}}, t\in\mathbb{T}$ then ...ha) f^{\Delta^k}(\alpha) + \displaystyle\int_{\alpha}^{\rho^{n-1}(t)} h_{n-1}(t,\sigma(\tau)) f^{\Delta^n}(\tau) \Delta \tau,$$
  516 bytes (83 words) - 17:05, 15 January 2023
• Delta cosh
  | $\cosh_1(t,0)=\dfrac{1}{2}\left( (1-h)^{\frac{t}{h}} + (1+h)^{\frac{t}{h}}\right) = \displaystyle\sum_{k=0}^{\infty} h_{2k}(t,0) $ |[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q &gt; 1$]]
  1 KB (169 words) - 14:13, 28 January 2023
• Joint time scales probability density function
  ...isplaystyle\int_0^{\infty} \int_0^{\infty} f_{X,Y}(x,y) \Delta y \Delta x=1$. ...umsystem.edu/xmlui/bitstream/handle/10355/29595/Matthews_2011.pdf?sequence=1 Probability theory on time scales and applications to finance and inequalit
  500 bytes (83 words) - 04:37, 6 March 2015
• Inverse cylinder transformation
  $$\xi_h^{-1}(z)=\dfrac{e^{zh}-1}{h}.$$
  552 bytes (86 words) - 00:57, 30 May 2017
• Abel dynamic equation
  ...$f_i,g_k \colon \mathbb{R} \rightarrow \mathbb{R}$ for $i=0,1,2$ and $k=0,1$. The Abel dynamic equation of the second kind is
  447 bytes (76 words) - 19:28, 5 April 2015
• Derivative of Delta sine
  \sin^{\Delta}_p(t,t_0) &= \dfrac{1}{2i} \dfrac{\Delta}{\Delta t} \left( e_{ip}(t,t_0) - e_{-ip}(t,t_0) \right) &= \dfrac{1}{2} (e_{ip}(t,t_0)+e_{-ip}(t,t_0)) \\
  601 bytes (104 words) - 21:28, 9 June 2016
• Derivative of delta cosine
  \cos_p^{\Delta}(t,t_0) &= \dfrac{1}{2} \dfrac{\Delta}{\Delta t} \Big(e_{ip}(t,t_0) + e_{-ip}(t,t_0) \Big) \\
  620 bytes (103 words) - 01:51, 6 February 2023
• Diamond alpha derivative
  Let $\mathbb{T}$ be a [[time scale]] and let $0\leq \alpha \leq 1$. The $\Diamond_{\alpha}$-derivative of a function $f \colon \mathbb{T} \ri $$\left| \alpha[f^{\sigma}(t)-f(s)]\eta_{ts} + (1-\alpha)[f^{\rho}(t)-f(s)]\mu_{ts}-f^{\Diamond_{\alpha}}\mu_{ts}\eta_{ts} \r
  2 KB (274 words) - 08:32, 12 April 2015
• Delta cosine
  where $i=\sqrt{-1}$.
  877 bytes (127 words) - 14:13, 28 January 2023
• Table:Time scale delta sine functions
  ...{\displaystyle\prod_{k=t_0}^{t-1}1+ip(k) - \displaystyle\prod_{k=t_0}^{t-1}1-ip(k)}{2i}$ |[[Quantum_q_greater_than_1 | $\overline{q^{\mathbb{Z}}}, q &gt; 1$]]
  597 bytes (86 words) - 18:39, 21 March 2015

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