Elliptic Integrals - 19.12 Asymptotic Approximations

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
19.12.E1 K ⁡ ( k ) = ∑ m = 0 ∞ ( 1 2 ) m ⁢ ( 1 2 ) m m ! ⁢ m ! ⁢ k ′ 2 ⁢ m ⁢ ( ln ⁡ ( 1 k ′ ) + d ⁢ ( m ) ) complete-elliptic-integral-first-kind-K 𝑘 superscript subscript 𝑚 0 Pochhammer 1 2 𝑚 Pochhammer 1 2 𝑚 𝑚 𝑚 superscript superscript 𝑘 ′ 2 𝑚 1 superscript 𝑘 ′ 𝑑 𝑚 {\displaystyle{\displaystyle K\left(k\right)=\sum_{m=0}^{\infty}\frac{{\left(% \tfrac{1}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}{k^{\prime}}^{% 2m}\left(\ln\left(\frac{1}{k^{\prime}}\right)+d(m)\right)}}
\compellintKk@{k} = \sum_{m=0}^{\infty}\frac{\Pochhammersym{\tfrac{1}{2}}{m}\Pochhammersym{\tfrac{1}{2}}{m}}{m!\;m!}{k^{\prime}}^{2m}\left(\ln@@{\left(\frac{1}{k^{\prime}}\right)}+d(m)\right)		 0 < | k ′ | , | k ′ | < 1 formulae-sequence 0 superscript 𝑘 ′ superscript 𝑘 ′ 1 {\displaystyle{\displaystyle 0<|k^{\prime}|,|k^{\prime}|<1}} 
EllipticK(k) = sum((pochhammer((1)/(2), m)*pochhammer((1)/(2), m))/(factorial(m)*factorial(m))*(sqrt(1 - (k)^(2)))^(2*m)*(ln((1)/(sqrt(1 - (k)^(2))))+ d(m)), m = 0..infinity)

EllipticK[(k)^2] == Sum[Divide[Pochhammer[Divide[1,2], m]*Pochhammer[Divide[1,2], m],(m)!*(m)!]*(Sqrt[1 - (k)^(2)])^(2*m)*(Log[Divide[1,Sqrt[1 - (k)^(2)]]]+ d[m]), {m, 0, Infinity}, GenerateConditions->None]
Failure Failure Error Skip - No test values generated
19.12.E2 E ⁡ ( k ) = 1 + 1 2 ⁢ ∑ m = 0 ∞ ( 1 2 ) m ⁢ ( 3 2 ) m ( 2 ) m ⁢ m ! ⁢ k ′ 2 ⁢ m + 2 ⁢ ( ln ⁡ ( 1 k ′ ) + d ⁢ ( m ) - 1 ( 2 ⁢ m + 1 ) ⁢ ( 2 ⁢ m + 2 ) ) complete-elliptic-integral-second-kind-E 𝑘 1 1 2 superscript subscript 𝑚 0 Pochhammer 1 2 𝑚 Pochhammer 3 2 𝑚 Pochhammer 2 𝑚 𝑚 superscript superscript 𝑘 ′ 2 𝑚 2 1 superscript 𝑘 ′ 𝑑 𝑚 1 2 𝑚 1 2 𝑚 2 {\displaystyle{\displaystyle E\left(k\right)=1+\frac{1}{2}\sum_{m=0}^{\infty}% \frac{{\left(\tfrac{1}{2}\right)_{m}}{\left(\tfrac{3}{2}\right)_{m}}}{{\left(2% \right)_{m}}m!}{k^{\prime}}^{2m+2}\*\left(\ln\left(\frac{1}{k^{\prime}}\right)% +d(m)-\frac{1}{(2m+1)(2m+2)}\right)}}
\compellintEk@{k} = 1+\frac{1}{2}\sum_{m=0}^{\infty}\frac{\Pochhammersym{\tfrac{1}{2}}{m}\Pochhammersym{\tfrac{3}{2}}{m}}{\Pochhammersym{2}{m}m!}{k^{\prime}}^{2m+2}\*\left(\ln@@{\left(\frac{1}{k^{\prime}}\right)}+d(m)-\frac{1}{(2m+1)(2m+2)}\right)		 | k ′ | < 1 superscript 𝑘 ′ 1 {\displaystyle{\displaystyle|k^{\prime}|<1}} 
EllipticE(k) = 1 +(1)/(2)*sum((pochhammer((1)/(2), m)*pochhammer((3)/(2), m))/(pochhammer(2, m)*factorial(m))*(sqrt(1 - (k)^(2)))^(2*m + 2)*(ln((1)/(sqrt(1 - (k)^(2))))+ d(m)-(1)/((2*m + 1)*(2*m + 2))), m = 0..infinity)

EllipticE[(k)^2] == 1 +Divide[1,2]*Sum[Divide[Pochhammer[Divide[1,2], m]*Pochhammer[Divide[3,2], m],Pochhammer[2, m]*(m)!]*(Sqrt[1 - (k)^(2)])^(2*m + 2)*(Log[Divide[1,Sqrt[1 - (k)^(2)]]]+ d[m]-Divide[1,(2*m + 1)*(2*m + 2)]), {m, 0, Infinity}, GenerateConditions->None]
Error Failure -
Failed [10 / 10]
Result: Indeterminate
Test Values: {Rule[d, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[k, 1]}


Result: Indeterminate
Test Values: {Rule[d, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]], Rule[k, 1]}

... skip entries to safe data
19.12#Ex1 d ⁢ ( m ) = ψ ⁡ ( 1 + m ) - ψ ⁡ ( 1 2 + m ) 𝑑 𝑚 digamma 1 𝑚 digamma 1 2 𝑚 {\displaystyle{\displaystyle d(m)=\psi\left(1+m\right)-\psi\left(\tfrac{1}{2}+% m\right)}}
d(m) = \digamma@{1+m}-\digamma@{\tfrac{1}{2}+m}		 

d(m) = Psi(1 + m)- Psi((1)/(2)+ m)

d[m] == PolyGamma[1 + m]- PolyGamma[Divide[1,2]+ m]
Failure Failure
Failed [30 / 30]
Result: .4797310429+.5000000000*I
Test Values: {d = 1/2*3^(1/2)+1/2*I, m = 1}


Result: 1.512423114+1.*I
Test Values: {d = 1/2*3^(1/2)+1/2*I, m = 2}

... skip entries to safe data
Failed [30 / 30]
Result: Complex[0.04671834077232884, 0.24999999999999997]
Test Values: {Rule[d, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[m, 1]}


Result: Complex[0.6463977093312149, 0.49999999999999994]
Test Values: {Rule[d, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[m, 2]}

... skip entries to safe data
19.12#Ex2 d ⁢ ( m + 1 ) = d ⁢ ( m ) - 2 ( 2 ⁢ m + 1 ) ⁢ ( 2 ⁢ m + 2 ) 𝑑 𝑚 1 𝑑 𝑚 2 2 𝑚 1 2 𝑚 2 {\displaystyle{\displaystyle d(m+1)=d(m)-\frac{2}{(2m+1)(2m+2)}}}
d(m+1) = d(m)-\frac{2}{(2m+1)(2m+2)}		 

d(m + 1) = d(m)-(2)/((2*m + 1)*(2*m + 2))
 
d[m + 1] == d[m]-Divide[2,(2*m + 1)*(2*m + 2)]
 Skipped - no semantic math Skipped - no semantic math - -
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