Equations in MathMl

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$H = \int_{0}^{\infty} F (λ) d λ$

$H = \sum_{i} F (λ) Δ λ$

$\begin{matrix} Δ λ = \frac{λ_{i + 1} + λ_{i}}{2} - \frac{λ_{i} + λ_{i - 1}}{2} \\ = \frac{λ_{i + 1} - λ_{i - 1}}{2} \end{matrix}$

$H_{i} = Δ λ \cdot F (λ_{i})$

$LSTM = 15^{0} ∆ T_{U T C}$

$E o T = 9.87 \sin (2 B) - 7.53 \cos (B) - 1.5 \sin (B)$

$α = 90 + φ - δ$

$S u n r i s e = 12 - \frac{1}{15^{0}} \cos^{- 1} (\frac{- \sin φ \sin δ}{\cos φ \cos δ})$

$S u n s e t = 12 + \frac{1}{15^{0}} \cos^{- 1} (\frac{- \sin φ \sin δ}{\cos φ \cos δ})$

$I_{D} = 1.353 \times {0.7}^{(A M^{0.678})}$

$A M = \frac{1}{\cos θ}$

$n_{i} (T) = 5.29 \times 10^{19} {(T / 300)}^{2.54} \exp (- 6726 / T)$

$\hat{E}$

$\vec{E}$

$\frac{d n}{d t} = \frac{d p}{d t} = 0$

$\frac{1}{q} \frac{d J_{n}}{d x} = (U - G) = 0$

$\frac{d E}{d x} = \frac{ρ}{ε} = \frac{q}{ε} (- N_{A} + N_{D})$

$\begin{matrix} \frac{d J_{T}}{d x} & = \frac{d (J_{p} + J_{n})}{d x} \\ = \frac{d J_{p}}{d x} + \frac{d J_{n}}{d x} \\ = - q (U_{p} + G_{p}) + q (U_{n} + G_{n}) \\ = q (G_{n} - G_{p}) - q (U_{n} - U_{p}) \end{matrix}$

$φ_{s u n} (E_{G}, \infty, 0, T_{s u n}) = \frac{2 π}{h^{3} c^{2}} \int_{E_{G}}^{\infty} \frac{E^{2}}{e x p (\frac{E}{k T_{s u n}}) - 1} d E$

$φ_{2} (E_{G}, \infty, μ, T_{E a r t h}) = \frac{2 π}{h^{3} c^{2}} \int_{E_{G}}^{\infty} \frac{E^{2}}{e x p (\frac{E - μ}{k T_{E a r t h}}) - 1} d E$

$I = I_{L} - I_{0} \exp [\frac{q (V + I R_{S})}{n k T}]$

$I = I_{L} - I_{0} \exp [\frac{q V}{n k T}] - \frac{V}{R_{S H}}$

$P_{M P}^{'} \approx V_{M P} I_{M P} - \frac{V_{M P}^{2}}{R_{S h}} = V_{M P} I_{M P} (1 - \frac{V_{M P}}{I_{M P}} \frac{1}{R_{S H}}) = P_{M P} (1 - \frac{V_{O C}}{I_{S C}} \frac{1}{R_{S H}})$

$V_{1} = V_{2}$

$I_{T} = I_{1} + I_{2}$

$J = J_{L} - J_{01} {e x p [\frac{q (V + J R_{s})}{k T}] - 1} - J_{02} {e x p [\frac{q (V + J R_{s})}{2 k T}] - 1} - \frac{V + J R_{s}}{R_{s h u n t}}$

$J = J_{01} {e x p [\frac{q (V - J R_{s})}{k T}] - 1} + J_{02} {e x p [\frac{q (V - J R_{s})}{2 k T}] - 1} + \frac{V - J R_{s}}{R_{s h u n t}}$

$J = J_{L} - J_{01} e x p [\frac{q (V + J R_{s})}{k T}] - J_{02} e x p [\frac{q (V + J R_{s})}{2 k T}] - \frac{V + J R_{s}}{R_{s h u n t}}$

$J = J_{01} e x p [\frac{q (V - J R_{s})}{k T}] + J_{02} e x p [\frac{q (V - J R_{s})}{2 k T}] + \frac{V - J R_{s}}{R_{s h u n t}}$

$\frac{π}{\ln (2)} = 4.532$

$ρ = \frac{π}{\ln (2)} (\frac{V}{I}) t = 4.532 (\frac{V}{I}) t$

$E (e V) = \frac{1.24}{λ (μ m)}$

$I = I_{L} - I_{0} \exp [\frac{q (V + I R_{S})}{n k T}]$

$Implied V_{o c} = \frac{k T}{q} \ln (\frac{Δ n (Δ n + N_{A})}{n_{i}^{2}})$

$r_{1} = \frac{n_{0} - n_{1}}{n_{0} + n_{1}}$

$r_{2} = \frac{n_{1} - n_{2}}{n_{1} + n_{2}}$

$r_{3} = \frac{n_{2} - n_{3}}{n_{2} + n_{3}}$

$θ_{1} = \frac{2 π n_{1} t_{1}}{λ}$

$θ_{2} = \frac{2 π n_{2} t_{2}}{λ}$

$R = | r^{2} | = \frac{r_{1}^{2} + r_{2}^{2} + r_{3}^{2} + r_{1}^{2} r_{2}^{2} r_{3}^{2} + 2 r_{1} r_{2} (1 + r_{3}^{2}) \cos 2 θ_{1} + 2 r_{2} r_{3} (1 + r_{1}^{2}) \cos 2 θ_{2} + 2 r_{1} r_{3} \cos 2 (θ_{1} + θ_{2}) + 2 r_{1} r_{2}^{2} r_{3} \cos 2 (θ_{1} - θ_{2})}{1 + r_{1}^{2} r_{2}^{2} + r_{1}^{2} r_{3}^{2} + r_{2}^{2} r_{3}^{2} + 2 r_{1} r_{2} (1 + r_{3}^{2}) \cos 2 θ_{1} + 2 r_{2} r_{3} (1 + r_{1}^{2}) \cos 2 θ + 2 r_{1} r_{3} \cos 2 (θ_{1} + θ_{2}) + 2 r_{1} r_{2}^{2} r_{3} \cos 2 (θ_{1} - θ_{2})}$

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

$\cos (θ + φ) = \cos (θ) \cos (φ) - \sin (θ) \sin (φ)$

$F (λ) = \frac{2 π h c^{2}}{λ^{5} (\exp (\frac{h c}{k λ T}) - 1)}$

$I_{D} = 1.353 \times {0.7}^{(A M^{0.678})}$

$R = {(\frac{n_{0} - n_{S i}}{n_{0} + n_{S i}})}^{2}$

$F (λ) = Φ E \frac{1}{Δ λ} in SI units$

$F (λ) = Φ q \frac{1.24}{λ (μ m)} \frac{1}{Δ λ (μ m)}$

$I = I_{L} - I_{0} [\exp (\frac{q V}{n k T})]$

$R_{b} = \frac{ρ l}{A} = \frac{ρ_{b} W}{A}$

$ρ_{□} = \frac{ρ}{t}$

$α = \sin^{- 1} [\sin δ \sin φ + \cos δ \cos φ \cos (H R A)]$

$ζ = 90^{0} - α$

$S u n r i s e = 12 - \frac{1}{15^{0}} \cos^{- 1} (\frac{- \sin φ \sin δ}{\cos φ \cos δ}) - \frac{T C}{60}$

$S u n s e t = 12 + \frac{1}{15^{0}} \cos^{- 1} (\frac{- \sin φ \sin δ}{\cos φ \cos δ}) - \frac{T C}{60}$

$S u n r i s e = 12 - \frac{1}{15^{0}} \cos^{- 1} (- \tan φ \tan δ) - \frac{T C}{60}$

$S u n s e t = 12 + \frac{1}{15^{0}} \cos^{- 1} (- \tan φ \tan δ) - \frac{T C}{60}$

$n_{0} p_{0} = n_{i}^{2}$

$\frac{1}{τ_{b u l k}} = \frac{1}{τ_{b a n d}} + \frac{1}{τ_{A u g e r}} + \frac{1}{τ_{S R H}}$

$L = \sqrt{D τ}$

$V_{O C} = \frac{k T}{q} \ln (\frac{(N_{A} + Δ n) Δ n}{n_{i}^{2}}$

$F F = \frac{V_{M P} I_{M P}}{V_{O C} I_{S C}}$

$J = J_{L} - J_{01} e x p (\frac{q (V + J R s)}{k T} - 1) - J_{02} e x p (\frac{q (V + J R s)}{n k T} - 1) - (\frac{V + J R_{s}}{R_{s h u n t}})$

$R_{⊥} = \frac{\sin^{2} (θ_{1} - θ_{2})}{\sin^{2} (θ_{1} + θ_{2})}$

$R_{∥} = \frac{\tan^{2} (θ_{1} - θ_{2})}{\tan^{2} (θ_{1} + θ_{2})}$

$R_{T} = \frac{R_{⊥} + R_{∥}}{2}$

$n_{1} \sin θ_{1} = n_{2} \sin θ_{2}$

$θ_{2} = \sin^{- 1} (\frac{n_{2}}{n_{1}} \sin θ_{1})$

$I = I_{L} - I_{0} [\exp (\frac{q V}{n k T}) - 1]$

$\frac{d E}{d x} = \frac{ρ}{ε} = \frac{q}{ε} (p (x) - n (x) - N_{A}^{-} + N_{D}^{+})$

$J_{n} = q μ_{n} n E + q D_{n} \frac{d n}{d x}, J_{p} = q μ_{p} p E + q D_{p} \frac{d p}{d x}$

$\frac{d E}{d x} = \frac{ρ}{ε}$

$W = x_{p} + x_{n} = \sqrt{\frac{2 ε}{q} V_{o} (\frac{1}{N_{A}} + \frac{1}{N_{D}})}$

$\ln (I) = \ln (I_{0}) + (\frac{q}{n k T})$

$G = α N_{0} e^{- α x}$

$E = E^{0} - \frac{R T}{n F} \times \ln (Q)$

$Q = \frac{{(D}^{d} {(C}^{c}}{{(A}^{a} {(B}^{b}}$

$θ_{1} = \sin^{- 1} (\frac{η_{2}}{η_{1}})$

$L i^{+} + e^{-} \to L i$

$L i \to L i^{+} + e^{-}$

$C l + e^{-} \to C l^{-}$

$2 H^{+} (a q) + 2 e^{-} \to H_{2} (g)$

$Z n^{2 +} (a q) + 2 e^{-} \to Z n (s) E ° = - 0.76 V$

$Z n (s) \to Z n^{2 +} (a q) + 2 e^{-} E ° = 0.76 V$

$C u^{2 +} (a q) + 2 e^{-} \to C u (s) E ° = 0.34 V$

$a A + b B \to c C + d D$

$\overset{V a l e n c e c h a r g e 0}{Z n (s)} \to \overset{2 +}{Z n^{2 +}} (a q) + 2 e^{-}$

$\overset{V a l e n c e c h a r g e 2 +}{C u^{2 +}} (a q) + 2 e^{-} \to \overset{0}{C u} (s)$

$C u^{2 +} (a q) + 2 e^{-} + Z n (s) \to C u (s) + Z n^{2 +} (a q) + 2 e^{-}$

$C u^{2 +} (a q) + Z n (s) \to C u (s) + Z n^{2 +} (a q)$

$C a p a c i t y (A h) = n \times F \times \frac{1 h o u r}{3600 s e c}$

$E n e r g y C a p a c i t y = A h \times B a t t e r y V o l t a g e$

$P b O_{2} + P b + 2 H_{2} S O_{4} \Leftrightarrow_{c h a r g e}^{d i s c h a r g e} 2 P b S O_{4} + 2 H_{2} O$

$P b + S {O_{4}}^{2 -} \Leftrightarrow_{c h a r g e}^{d i s c h a r g e} P b S O_{4} + 2 e^{-}$

$P b O_{2} + S {O_{4}}^{2 -} + 4 H^{+} + 2 e^{-} \Leftrightarrow_{c h a r g e}^{d i s c h a r g e} P b S O_{4} + 2 H_{2} O$

$Z n (s) + 2 O H^{-} \to Z n {(O H)}_{2} + 2 e^{-} (a q) M n O_{2} (s) + H_{2} O + 2 e^{-} \to M n_{2} O_{3} (s) + 2 O H^{-} (a q)$

$A M = \frac{1}{\cos (θ)}$

$A M = \frac{1}{\cos (θ) + 0.50572 {(96.07995 - θ)}^{- 1.6364}}$

$A M = \sqrt{1 + {(\frac{s}{h})}^{2}}$

$I_{D} = 1.353 \cdot 0 . 7^{A M^{0.678}}$

$I_{D} = 1.353 \cdot ((1 - a h) 0 . 7^{A M^{0.678}} + a h$

$I_{G} = 1.1 \cdot I_{D}$

$S_{m o d u l e} = S_{i n c i d e n t} \cos (α) \sin (β) \cos (Ψ - Θ) + \sin (α) \cos (β))$

$S_{m o d u l e} = S_{i n c i d e n t} \cos (Υ) = S_{i n c i d e n t} S \cdot N$

$F (λ) = \frac{2 π h c^{2}}{λ^{5} (e x p (\frac{h c}{k λ T}) - 1)}$

$H = σ T^{4}$

$λ_{ρ} (μ m) = \frac{2900}{T}$

$E = \frac{h c}{λ}$

$E (e V) = \frac{1.24}{λ (μ m)}$

$\begin{matrix} H (\frac{W}{m^{2}}) = Φ \times \frac{h c}{λ} using SI units \\ H (\frac{W}{m^{2}}) = Φ \times q \frac{1.24}{λ (μ m)} for wavelength in μm \\ H (\frac{W}{m^{2}}) = Φ \times q E (e V) for energy in eV \end{matrix}$

$d_{1} = \frac{λ_{0}}{4 η_{1}}$

$n_{1} = \sqrt{n_{0} n_{2}}$

$r_{1} = \frac{n_{0} - n_{1}}{n_{0} + n_{1}}$

$r_{2} = \frac{n_{1} - n_{2}}{n_{1} + n_{2}}$

$θ = \frac{2 π n_{1} t_{1}}{λ}$

$R = | r^{2} | = \frac{r_{1}^{2} + r_{2}^{2} + 2 r_{1} r_{2} \cos 2 θ}{1 + r_{1}^{2} r_{2}^{2} + 2 r_{1} r_{2} \cos 2 θ}$

$δ = - 23.45 ° \times \cos (\frac{360}{365} \times (d + 10))$

$H_{0} = \frac{R_{s u n}^{2}}{D^{2}} H_{s u n}$

$B = \frac{360}{365} (d) (- 81)$

$α = 90 - φ + δ$

$α = 90 + φ - δ$

$Amanecer = 12 - \frac{1}{15^{0}} \cos^{- 1} (\frac{- \sin φ \sin δ}{\cos φ \cos δ})$

$P d s = 12 + \frac{1}{15^{0}} \cos^{- 1} (\frac{- \sin φ \sin δ}{\cos φ \cos δ})$

$v_{o c} = \frac{q}{n k T} V_{o c}$

$F F = \frac{v_{o c} - \ln (v_{o c} + 0.72)}{v_{o c} + 1}$

$S R (A/W) = \frac{Q E \cdot λ (nm)}{1239.8}$

$S R (A/W) = \frac{Q E \cdot λ (μ m)}{1.2398}$

$I_{0} = q A \frac{D n_{i}^{2}}{L N_{D}}$

$R_{C H} = \frac{V_{M P}}{I_{M P}} \approx \frac{V_{O C}}{I_{S C}}$

$δ = 23.45 ° \times \sin (\frac{360}{365} \times (d + 284))$

$δ = 23.45 ° \times \sin (\frac{360}{365} \times (d - 81))$

$σ = q (μ_{n} n + μ_{p} p)$

$I = I_{L} - I_{0} e^{\frac{V}{V_{t}}}$

$V_{t} = \frac{n k T}{q}$

$P = V I_{L} - V I_{0} e^{\frac{V}{V_{t}}}$

$u = V I_{0}$

$u^{'} = I_{0}$

$v = e^{\frac{V}{V_{t}}}$

$v^{'} = \frac{1}{V_{t}} e^{\frac{V}{V_{t}}}$

$\frac{d P}{d V} = I_{L} - (V I_{0} \frac{1}{V_{t}} e^{\frac{V}{V_{t}}} + I_{0} e^{\frac{V}{V_{t}}}$

$\frac{d P}{d V} = 0$

$V_{m p} I_{0} \frac{1}{V_{t}} e^{\frac{V_{m p}}{V_{t}}} - I_{0} e^{\frac{V_{m p}}{V_{t}}} = I_{L}$

$\frac{V_{m p}}{V_{t}} e^{\frac{V_{m p}}{V_{t}}} - e^{\frac{V_{m p}}{V_{t}}} = \frac{I_{L}}{I_{0}}$

$e^{\frac{V_{m p}}{V_{t}}} (\frac{V_{m p}}{V_{t}} - 1) = \frac{I_{L}}{I_{0}}$

$\frac{V_{m p}}{V_{t}} + \ln (\frac{V_{m p}}{V_{t}} - 1) = l n (\frac{I_{L}}{I_{0}})$

$V_{m p} = V_{t} l n (\frac{I_{L}}{I_{0}}) - \ln (\frac{V_{m p}}{V_{t}} - 1)$

$V_{o c} = V_{t} l n (\frac{I_{L}}{I_{0}})$

$V_{m p} = V_{o c} - \ln (\frac{V_{m p}}{V_{t}} - 1)$

$0 = I_{L} - I_{0} \exp (\frac{V}{V_{t}}) (1 + \frac{V}{V_{t}})$

$\frac{I_{L}}{I_{0}} = \exp (\frac{V}{V_{t}}) \times (\frac{V}{V_{t}})$

$Y = X e^{X} \Leftrightarrow X = W (Y). Using this relation the previous equatoin becomes:$

$\frac{V}{V_{t}} = W (\frac{J_{L}}{J_{0}})$

$V_{m p} = V_{t} W (\frac{J_{L}}{J_{0}})$

$V_{m p} = V_{t} (W (e \frac{J_{L}}{J_{0}}) - 1$

$J = J_{L} - J_{0} [exp (\frac{V}{V_{t}}) - 1]$

$P = V J_{L} - V J_{0} [exp (\frac{V}{V_{t}}) - 1]$

$E_{G} (T) = E_{G} (0) - \frac{α T^{2}}{T + β}$

$E_{G} (T) = E_{B} - a_{B} [1 + \frac{2}{(e^{\frac{Θ}{T}} - 1)}]$

$E_{G} (T) = E_{G} (0) + B T + C T^{2}$

$E_{G} (T) = E_{B} (0) - \frac{α Θ}{2} [\sqrt[p]{1 + {(\frac{2 T}{Θ})}^{p}} - 1]$

$E_{G}^{AB} = {xE}_{G}^{B} + E_{G}^{A} (1 - x) - C (1 - x) x$

$E_{G, X}^{AB} = {xE}_{G, X}^{B} + E_{G, X}^{A} (1 - x) - C_{X} (1 - x) x$

$E_{G, L}^{AB} = {xE}_{G, L}^{B} + E_{G, L}^{A} (1 - x) - C_{L} (1 - x) x$

$E_{G, Γ}^{AB} = {xE}_{G, Γ}^{B} + E_{G, Γ}^{A} (1 - x) - C_{Γ} (1 - x) x$

$a_{}^{AB} = {xa}_{}^{B} + a_{}^{A} (1 - x) - C (1 - x) x$

${CBO}_{AB} = χ_{A} - χ_{B}$

$m_{e, DOS}^{*} = M_{C}^{\frac{2}{3}} {({m_{l}^{*} m_{t}^{*} m}_{t}^{*})}^{\frac{1}{3}}$

$m_{e, cond}^{*} = \frac{3}{(\frac{1}{m_{l}^{*}} + \frac{1}{m_{t}^{*}} + \frac{1}{m_{t}^{*}})}$

$m_{h, DOS}^{*} = {[{m_{hh}^{*}}^{\frac{3}{2}} + {m_{lh}^{*}}^{\frac{3}{2}} + {(m_{so}^{*} e^{\frac{- Δ}{kT}})}^{\frac{3}{2}}]}^{\frac{2}{3}}$

$m_{h, cond}^{*} = \frac{3}{(\frac{1}{m_{hh}^{*}} + \frac{1}{m_{lh}^{*}} + \frac{1}{m_{so}^{*}})}$

$N_{V} = 2 {(\frac{2 π m_{h}^{*} k_{B} T}{h^{2}})}^{\frac{3}{2}}$

$N_{C} = 2 {(\frac{2 π m_{e}^{*} k_{B} T}{h^{2}})}^{\frac{3}{2}}$

$n_{i}^{2} = N_{C} N_{V} e^{\frac{- E_{G}}{kT}}$

$n_{i} = 9.15 \times 10^{10} {(\frac{T}{300})}^{2} e^{\frac{- 0.5928}{kT}}$

$v_{d} = μE$

$μ = \frac{q \bar{τ}}{m^{*}}$

$μ = μ_{min} + \frac{μ_{max} - μ_{min}}{1 + {(\frac{N}{N_{r}})}^{α}}$

$\frac{1}{τ_{eff}} = \frac{1}{τ_{Auger}} + \frac{1}{τ_{radiative}} + \frac{1}{τ_{SRH}} + \frac{1}{τ_{Surf}}$

$U_{rad} = B n p$

$U \equiv \frac{Δ n}{τ_{}}$

$U_{rad} = B (n p - n_{i}^{2})$

$U_{rad} \equiv \frac{Δ n}{τ_{Rad}} = B (n p - n_{i}^{2})$

$τ_{Rad} = \frac{1}{B ((n_{0} + p_{0}) + Δ n)}$

$τ_{Rad} = \frac{1}{B N_{D}^{}}$

$τ_{Rad} = \frac{1}{B N_{A}^{}}$

$U_{Auger} = C_{n} n (np - n_{i}^{2}) + C_{p} p (np - n_{i}^{2})$

$U_{Auger} ≅ C_{n} n^{2} p + C_{p} p^{2} n$

$∖ n U_{Auger} \equiv \frac{Δ n}{τ_{Auger}}$

$τ_{Auger} = \frac{1}{C_{p} ({n_{0}^{2} + 2 n}_{0} Δ n + {Δ n}^{2}) + C_{n} ({p_{0}^{2} + 2 p}_{0} Δ n + {Δ n}^{2})}$

$τ_{Auger} = \frac{1}{C_{p} N_{D}^{2}}$

$τ_{Auger} = \frac{1}{C_{n} N_{A}^{2}}$

$\frac{Δ n}{τ_{Rad}} = B ((n_{0} + Δ n) (p_{0} + Δ n) - n_{i}^{2})$

$\frac{Δ n}{τ_{Rad}} = B (n_{0} p_{0} + Δ n (n_{0} + p_{0}) + Δ n 2 - n_{i}^{2})$

$\frac{Δ n}{τ_{Rad}} = B (Δ n (n_{0} + p_{0}) + Δ n 2)$

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Equations in MathMl