about the minimax and orthogonal polynomial Exponential function
using legendre Polynomial between 0,1 for the exponential function you could get te following series
$$e^{-frac{p}{2}}=sum _{n=0}^{infty } sqrt{frac{pi }{2}} (-1)^{ n} (2 n+1) I_{n+frac{1}{2}}(1) P_nleft(frac{p}{2}right)$$ it is suppose that t is a good aproximation using only m=2 you get the error
$$e^{-frac{p}{2}}simeq -frac{5}{8} sqrt{frac{pi }{2}} left(3 p^2-4right) I_{frac{5}{2}}(1)+frac{3}{2} sqrt{frac{pi }{2}} p I_{frac{3}{2}}(1)+left(-sqrt{frac{pi }{2}}right) I_{frac{1}{2}}(1)$$ for p=1 the error it is about
0.027875
it is suppose that this series could be improve ( maybe get it better improve using chevyshef series) but de following series
$$e^{-frac{p}{2}}simeq frac{3 pi (4 p+7) I_{frac{3}{4}}left(frac{p}{2}right)}{8 p^{3/4} Gamma left(frac{1}{4}right)}-frac{15 pi sqrt[4]{p} I_{-frac{1}{4}}left(frac{p}{2}right)}{8 Gamma left(frac{1}{4}right)}$$
for m=2 (usin two terms) gives te error -0.00286246 10 times better
how it is possible? it is suppose that Legendre it is orthogonal polinomial
calculus
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using legendre Polynomial between 0,1 for the exponential function you could get te following series
$$e^{-frac{p}{2}}=sum _{n=0}^{infty } sqrt{frac{pi }{2}} (-1)^{ n} (2 n+1) I_{n+frac{1}{2}}(1) P_nleft(frac{p}{2}right)$$ it is suppose that t is a good aproximation using only m=2 you get the error
$$e^{-frac{p}{2}}simeq -frac{5}{8} sqrt{frac{pi }{2}} left(3 p^2-4right) I_{frac{5}{2}}(1)+frac{3}{2} sqrt{frac{pi }{2}} p I_{frac{3}{2}}(1)+left(-sqrt{frac{pi }{2}}right) I_{frac{1}{2}}(1)$$ for p=1 the error it is about
0.027875
it is suppose that this series could be improve ( maybe get it better improve using chevyshef series) but de following series
$$e^{-frac{p}{2}}simeq frac{3 pi (4 p+7) I_{frac{3}{4}}left(frac{p}{2}right)}{8 p^{3/4} Gamma left(frac{1}{4}right)}-frac{15 pi sqrt[4]{p} I_{-frac{1}{4}}left(frac{p}{2}right)}{8 Gamma left(frac{1}{4}right)}$$
for m=2 (usin two terms) gives te error -0.00286246 10 times better
how it is possible? it is suppose that Legendre it is orthogonal polinomial
calculus
asked yesterday
CLERKRAMA
13
New contributor
CLERKRAMA is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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using legendre Polynomial between 0,1 for the exponential function you could get te following series
$$e^{-frac{p}{2}}=sum _{n=0}^{infty } sqrt{frac{pi }{2}} (-1)^{ n} (2 n+1) I_{n+frac{1}{2}}(1) P_nleft(frac{p}{2}right)$$ it is suppose that t is a good aproximation using only m=2 you get the error
$$e^{-frac{p}{2}}simeq -frac{5}{8} sqrt{frac{pi }{2}} left(3 p^2-4right) I_{frac{5}{2}}(1)+frac{3}{2} sqrt{frac{pi }{2}} p I_{frac{3}{2}}(1)+left(-sqrt{frac{pi }{2}}right) I_{frac{1}{2}}(1)$$ for p=1 the error it is about
0.027875
it is suppose that this series could be improve ( maybe get it better improve using chevyshef series) but de following series
$$e^{-frac{p}{2}}simeq frac{3 pi (4 p+7) I_{frac{3}{4}}left(frac{p}{2}right)}{8 p^{3/4} Gamma left(frac{1}{4}right)}-frac{15 pi sqrt[4]{p} I_{-frac{1}{4}}left(frac{p}{2}right)}{8 Gamma left(frac{1}{4}right)}$$
for m=2 (usin two terms) gives te error -0.00286246 10 times better
how it is possible? it is suppose that Legendre it is orthogonal polinomial
calculus
asked yesterday
CLERKRAMA
13
New contributor
CLERKRAMA is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
using legendre Polynomial between 0,1 for the exponential function you could get te following series
$$e^{-frac{p}{2}}=sum _{n=0}^{infty } sqrt{frac{pi }{2}} (-1)^{ n} (2 n+1) I_{n+frac{1}{2}}(1) P_nleft(frac{p}{2}right)$$ it is suppose that t is a good aproximation using only m=2 you get the error
$$e^{-frac{p}{2}}simeq -frac{5}{8} sqrt{frac{pi }{2}} left(3 p^2-4right) I_{frac{5}{2}}(1)+frac{3}{2} sqrt{frac{pi }{2}} p I_{frac{3}{2}}(1)+left(-sqrt{frac{pi }{2}}right) I_{frac{1}{2}}(1)$$ for p=1 the error it is about
0.027875
it is suppose that this series could be improve ( maybe get it better improve using chevyshef series) but de following series
$$e^{-frac{p}{2}}simeq frac{3 pi (4 p+7) I_{frac{3}{4}}left(frac{p}{2}right)}{8 p^{3/4} Gamma left(frac{1}{4}right)}-frac{15 pi sqrt[4]{p} I_{-frac{1}{4}}left(frac{p}{2}right)}{8 Gamma left(frac{1}{4}right)}$$
for m=2 (usin two terms) gives te error -0.00286246 10 times better
how it is possible? it is suppose that Legendre it is orthogonal polinomial
calculus
calculus
asked yesterday
CLERKRAMA
13
New contributor
CLERKRAMA is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
CLERKRAMA
13
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asked yesterday
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