User contributions for Robkahn131

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01:0101:01, 1 April 2025 diff hist +72‎ m Arccosine Function in terms of Gaussian Hypergeometric Function ‎No edit summary current
01:0001:00, 1 April 2025 diff hist +156‎ m Arctangent Function in terms of Gaussian Hypergeometric Function ‎No edit summary
00:5900:59, 1 April 2025 diff hist +158‎ m Arcsine Function in terms of Gaussian Hypergeometric Function ‎No edit summary

18:2318:23, 30 March 2025 diff hist +3,294‎ General Solution to Chebyshev's Differential Equation ‎ going out on a limb here - this appears to be an even more general solution than what I see published current
18:1218:12, 30 March 2025 diff hist +22‎ m Linear Second Order ODE/y'' + k^2 y = 0/Proof 1 ‎No edit summary current

16:0016:00, 29 March 2025 diff hist +5‎ m Linear Combination of Solutions to Homogeneous Linear 2nd Order ODE ‎No edit summary current
15:4115:41, 29 March 2025 diff hist +2,458‎ N General Solution to Chebyshev's Differential Equation/Lemma/Proof 2 ‎ Created page with "== Theorem == {{:General Solution to Chebyshev's Differential Equation/Lemma}} == Proof == <onlyinclude> Let: {{begin-eqn}} {{eqn | l = x | r = \sin \theta | c = }} {{eqn | ll= \leadsto | l = \d x | r = \cos \theta \rd \theta | c = }} {{eqn | ll= \leadsto | l = \frac {\d \theta} {\d x} | r = \frac 1 {\cos \theta} | c = }} {{eqn | ll= \leadsto | l = \frac {\d^2 \theta} {\d x \d \theta} | r = \frac {\sin \the..."
15:4115:41, 29 March 2025 diff hist +2,466‎ N General Solution to Chebyshev's Differential Equation/Lemma/Proof 1 ‎ Created page with "== Theorem == {{:General Solution to Chebyshev's Differential Equation/Lemma}} == Proof == <onlyinclude> Let: {{begin-eqn}} {{eqn | l = x | r = \cos \theta | c = }} {{eqn | ll= \leadsto | l = \d x | r = -\sin \theta \rd \theta | c = }} {{eqn | ll= \leadsto | l = \frac {\d \theta} {\d x} | r = -\frac 1 {\sin \theta} | c = }} {{eqn | ll= \leadsto | l = \frac {\d^2 \theta} {\d x \d \theta} | r = \frac {\cos \t..."
15:4115:41, 29 March 2025 diff hist +650‎ N General Solution to Chebyshev's Differential Equation/Lemma ‎ Created page with "== Theorem == <onlyinclude> '''Chebyshev's differential equation''': :$\quad \ds \paren {1 - x^2} \frac {\d^2 y} {\d x^2} - x \frac {\d y} {\d x} + n^2 y = 0$ can be depressed down to: :$\ds \frac {\d^2 y} {\d \theta^2} + n^2 y$ </onlyinclude> == Proof 1 == {{:General Solution to Chebyshev's Differential Equation/..."

20:1020:10, 28 March 2025 diff hist +77‎ m Sum of Logarithms/Complex Logarithm ‎No edit summary
20:0720:07, 28 March 2025 diff hist +108‎ m Sum of Logarithms ‎No edit summary
20:0620:06, 28 March 2025 diff hist +26‎ m Digamma Additive Formula ‎No edit summary current
20:0520:05, 28 March 2025 diff hist −53‎ m Digamma Additive Formula ‎No edit summary
20:0420:04, 28 March 2025 diff hist +1,820‎ N Sum of Logarithms/Complex Logarithm ‎ Created page with "== Theorem == <onlyinclude> Let $x, y \in \C$ where $x = r_1 e^{i \theta_1}$ and $y = r_2 e^{i \theta_2}$ such that $x \ne 0$ and $y \ne 0$. Then: :$\ln x + \ln y = \map \ln {x y}$ where $\ln$ denotes the complex natural logarithm. </onlyinclude> == Proof == We have: {{begin-eqn}} {{eqn | l = x | r = r_1 e^{i \theta_1} | c = }} {{eqn | ll = \leadsto | l = \ln x | r = \map \ln {r_1} + i \p..."
18:0618:06, 28 March 2025 diff hist +82‎ m Properties of Real Cosine Function ‎No edit summary current
18:0518:05, 28 March 2025 diff hist +80‎ m Properties of Real Sine Function ‎No edit summary current
18:0218:02, 28 March 2025 diff hist +233‎ m General Solution to Chebyshev's Differential Equation ‎No edit summary
17:4817:48, 28 March 2025 diff hist +212‎ N Talk:Properties of Real Sine Function ‎ Created page with "Shouldn't it be mentioned somewhere that the Real sine and Real cosine only map numbers to the closed interval [-1, 1]? --~~~~"
02:2902:29, 28 March 2025 diff hist +1,530‎ General Solution to Chebyshev's Differential Equation ‎No edit summary

21:3621:36, 27 March 2025 diff hist +620‎ m Talk:General Solution to Chebyshev's Differential Equation ‎No edit summary
21:2921:29, 27 March 2025 diff hist +973‎ Talk:General Solution to Chebyshev's Differential Equation ‎No edit summary
19:3519:35, 27 March 2025 diff hist +18‎ m Derivative of Solution to Constant Coefficient Homogeneous LSOODE is also Solution ‎No edit summary current
19:3419:34, 27 March 2025 diff hist +18‎ m Condition for Solutions to Constant Coefficient Homogeneous LSOODE to tend to Zero ‎No edit summary current
19:2419:24, 27 March 2025 diff hist +81‎ m Solution of Constant Coefficient Homogeneous LSOODE ‎No edit summary current
19:2119:21, 27 March 2025 diff hist +3,652‎ Talk:General Solution to Chebyshev's Differential Equation ‎No edit summary
17:2817:28, 27 March 2025 diff hist +112‎ m Sum of Logarithms/Natural Logarithm ‎No edit summary current
17:2717:27, 27 March 2025 diff hist +586‎ N Sum of Logarithms/Natural Logarithm/Proof 5 ‎ Created page with "== Theorem == {{:Sum of Logarithms/Natural Logarithm}} == Proof == <onlyinclude> {{begin-eqn}} {{eqn | l = x y | r = x \times y | c = }} {{eqn | ll = \leadsto | l = e^{\map \ln {xy} } | r = e^{\ln x} \times e^{\ln y} | c = Exponential of Natural Logarithm }} {{eqn | ll = \leadsto | l = e^{\map \ln {xy} } | r = e^{\ln x + \ln y} | c = Product of Powers }} {{eqn | ll = \leadsto | l = \map \ln {xy} | r =..." current
13:5813:58, 27 March 2025 diff hist +561‎ m General Solution to Chebyshev's Differential Equation ‎No edit summary
13:4413:44, 27 March 2025 diff hist +68‎ m Solution of Constant Coefficient Homogeneous LSOODE/Complex Roots of Auxiliary Equation ‎No edit summary current

21:5221:52, 26 March 2025 diff hist +4,415‎ General Solution to Chebyshev's Differential Equation ‎No edit summary
21:3921:39, 26 March 2025 diff hist +8‎ m Combination of Solutions to Non-Homogeneous LSOODE with same Homogeneous Part ‎No edit summary current
21:3721:37, 26 March 2025 diff hist +16‎ m Method of Variation of Parameters ‎No edit summary current
21:3321:33, 26 March 2025 diff hist +2‎ m Fundamental Solution to y'' + w y ‎No edit summary current
00:1200:12, 26 March 2025 diff hist +2,720‎ Legendre Polynomial in terms of Gaussian Hypergeometric Function ‎No edit summary

14:0714:07, 25 March 2025 diff hist +17‎ m Solution of Second Order Differential Equation with Missing Dependent Variable ‎No edit summary
14:0414:04, 25 March 2025 diff hist +34‎ m Solution of Second Order Differential Equation with Missing Dependent Variable ‎No edit summary
14:0314:03, 25 March 2025 diff hist +17‎ m Solution of Second Order Differential Equation with Missing Independent Variable ‎No edit summary
13:4413:44, 25 March 2025 diff hist +17‎ m Existence of Solution of 2nd Order Linear ODE ‎No edit summary current
13:4313:43, 25 March 2025 diff hist +17‎ m Existence of Solution to System of First Order ODEs ‎No edit summary current
00:2900:29, 25 March 2025 diff hist +4,554‎ N Chebyshev Polynomial of the First Kind in terms of Gaussian Hypergeometric Function/Proof 2 ‎ Created page with "== Theorem == {{:Chebyshev Polynomial of the First Kind in terms of Gaussian Hypergeometric Function}} == Proof == <onlyinclude> From De Moivre's Formula, we have: :$\cos n \theta + i \sin n \theta = \paren {\cos \theta + i \sin \theta}^n$ As $n \in \Z_{>0}$, we use the Binomial Theorem on the {{RHS}}, resulting in: :$\ds \cos n \theta + i \sin n \theta = \sum_{k \mathop \ge 0} \binom n k \paren {\cos^{n - k} \theta} \paren {i \sin \theta}^k$ When $k$ is De..." current
00:1300:13, 25 March 2025 diff hist +387‎ Chebyshev Polynomial of the First Kind in terms of Gaussian Hypergeometric Function ‎No edit summary current
00:1300:13, 25 March 2025 diff hist +3,645‎ N Chebyshev Polynomial of the First Kind in terms of Gaussian Hypergeometric Function/Proof 1 ‎ Created page with "== Theorem == {{:Chebyshev Polynomial of the First Kind in terms of Gaussian Hypergeometric Function}} == Proof == <onlyinclude> From Solution to Hypergeometric Differential Equation, we have: The '''hypergeometric series''': {{begin-eqn}} {{eqn | l = \map F {a, b, c; x} | r = \sum_{n \mathop = 0}^\infty \dfrac {a^{\overline n} b^{\overline n} } {c^{\overline n} \, n!} x^n }} {{end-eqn}} defines a Definition:Solution t..." current

01:0401:04, 22 March 2025 diff hist 0‎ m Power Series Expansion for Chebyshev Polynomial of the First Kind ‎No edit summary current
01:0401:04, 22 March 2025 diff hist +1,887‎ Power Series Expansion for Chebyshev Polynomial of the Second Kind ‎No edit summary

18:0518:05, 21 March 2025 diff hist −11‎ m Power Series Expansion for Chebyshev Polynomial of the First Kind ‎No edit summary
18:0518:05, 21 March 2025 diff hist 0‎ m Power Series Expansion for Chebyshev Polynomial of the First Kind ‎No edit summary
18:0218:02, 21 March 2025 diff hist +91‎ m Power Series Expansion for Chebyshev Polynomial of the First Kind ‎No edit summary
18:0118:01, 21 March 2025 diff hist −84‎ m Power Series Expansion for Chebyshev Polynomial of the First Kind ‎No edit summary
17:5717:57, 21 March 2025 diff hist +1,697‎ Power Series Expansion for Chebyshev Polynomial of the First Kind ‎No edit summary

16:2816:28, 9 March 2025 diff hist +118‎ Legendre's Duplication Formula ‎No edit summary current

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