Elliptic Integrals - 19.5 Maclaurin and Related Expansions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
19.5.E1 K ⁡ ( k ) = π 2 ⁢ ∑ m = 0 ∞ ( 1 2 ) m ⁢ ( 1 2 ) m m ! ⁢ m ! ⁢ k 2 ⁢ m complete-elliptic-integral-first-kind-K 𝑘 𝜋 2 superscript subscript 𝑚 0 Pochhammer 1 2 𝑚 Pochhammer 1 2 𝑚 𝑚 𝑚 superscript 𝑘 2 𝑚 {\displaystyle{\displaystyle K\left(k\right)=\frac{\pi}{2}\sum_{m=0}^{\infty}% \frac{{\left(\tfrac{1}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}k% ^{2m}}}
\compellintKk@{k} = \frac{\pi}{2}\sum_{m=0}^{\infty}\frac{\Pochhammersym{\tfrac{1}{2}}{m}\Pochhammersym{\tfrac{1}{2}}{m}}{m!\;m!}k^{2m}		 

EllipticK(k) = (Pi)/(2)*sum((pochhammer((1)/(2), m)*pochhammer((1)/(2), m))/(factorial(m)*factorial(m))*(k)^(2*m), m = 0..infinity)

EllipticK[(k)^2] == Divide[Pi,2]*Sum[Divide[Pochhammer[Divide[1,2], m]*Pochhammer[Divide[1,2], m],(m)!*(m)!]*(k)^(2*m), {m, 0, Infinity}, GenerateConditions->None]
Failure Successful Error Successful [Tested: 3]
19.5.E1 π 2 ⁢ ∑ m = 0 ∞ ( 1 2 ) m ⁢ ( 1 2 ) m m ! ⁢ m ! ⁢ k 2 ⁢ m = π 2 ⁢ F 1 2 ⁡ ( 1 2 , 1 2 ; 1 ; k 2 ) 𝜋 2 superscript subscript 𝑚 0 Pochhammer 1 2 𝑚 Pochhammer 1 2 𝑚 𝑚 𝑚 superscript 𝑘 2 𝑚 𝜋 2 Gauss-hypergeometric-F-as-2F1 1 2 1 2 1 superscript 𝑘 2 {\displaystyle{\displaystyle\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{{\left(% \tfrac{1}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}k^{2m}=\frac{% \pi}{2}{{}_{2}F_{1}}\left(\tfrac{1}{2},\tfrac{1}{2};1;k^{2}\right)}}
\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{\Pochhammersym{\tfrac{1}{2}}{m}\Pochhammersym{\tfrac{1}{2}}{m}}{m!\;m!}k^{2m} = \frac{\pi}{2}\genhyperF{2}{1}@{\tfrac{1}{2},\tfrac{1}{2}}{1}{k^{2}}		 

(Pi)/(2)*sum((pochhammer((1)/(2), m)*pochhammer((1)/(2), m))/(factorial(m)*factorial(m))*(k)^(2*m), m = 0..infinity) = (Pi)/(2)*hypergeom([(1)/(2),(1)/(2)], [1], (k)^(2))

Divide[Pi,2]*Sum[Divide[Pochhammer[Divide[1,2], m]*Pochhammer[Divide[1,2], m],(m)!*(m)!]*(k)^(2*m), {m, 0, Infinity}, GenerateConditions->None] == Divide[Pi,2]*HypergeometricPFQ[{Divide[1,2],Divide[1,2]}, {1}, (k)^(2)]
Failure Successful
Failed [3 / 3]
Result: Float(infinity)+Float(infinity)*I
Test Values: {k = 1}


Result: Float(infinity)+1.078257824*I
Test Values: {k = 2}

... skip entries to safe data
 Successful [Tested: 3]
19.5.E2 E ⁡ ( k ) = π 2 ⁢ ∑ m = 0 ∞ ( - 1 2 ) m ⁢ ( 1 2 ) m m ! ⁢ m ! ⁢ k 2 ⁢ m complete-elliptic-integral-second-kind-E 𝑘 𝜋 2 superscript subscript 𝑚 0 Pochhammer 1 2 𝑚 Pochhammer 1 2 𝑚 𝑚 𝑚 superscript 𝑘 2 𝑚 {\displaystyle{\displaystyle E\left(k\right)=\frac{\pi}{2}\sum_{m=0}^{\infty}% \frac{{\left(-\tfrac{1}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}% k^{2m}}}
\compellintEk@{k} = \frac{\pi}{2}\sum_{m=0}^{\infty}\frac{\Pochhammersym{-\tfrac{1}{2}}{m}\Pochhammersym{\tfrac{1}{2}}{m}}{m!\;m!}k^{2m}		 

EllipticE(k) = (Pi)/(2)*sum((pochhammer(-(1)/(2), m)*pochhammer((1)/(2), m))/(factorial(m)*factorial(m))*(k)^(2*m), m = 0..infinity)

EllipticE[(k)^2] == Divide[Pi,2]*Sum[Divide[Pochhammer[-Divide[1,2], m]*Pochhammer[Divide[1,2], m],(m)!*(m)!]*(k)^(2*m), {m, 0, Infinity}, GenerateConditions->None]
Failure Successful
Failed [2 / 3]
Result: Float(infinity)+1.343854231*I
Test Values: {k = 2}


Result: Float(infinity)+2.498348128*I
Test Values: {k = 3}

Successful [Tested: 3]
19.5.E2 π 2 ⁢ ∑ m = 0 ∞ ( - 1 2 ) m ⁢ ( 1 2 ) m m ! ⁢ m ! ⁢ k 2 ⁢ m = π 2 ⁢ F 1 2 ⁡ ( - 1 2 , 1 2 ; 1 ; k 2 ) 𝜋 2 superscript subscript 𝑚 0 Pochhammer 1 2 𝑚 Pochhammer 1 2 𝑚 𝑚 𝑚 superscript 𝑘 2 𝑚 𝜋 2 Gauss-hypergeometric-F-as-2F1 1 2 1 2 1 superscript 𝑘 2 {\displaystyle{\displaystyle\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{{\left(-% \tfrac{1}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}k^{2m}=\frac{% \pi}{2}{{}_{2}F_{1}}\left(-\tfrac{1}{2},\tfrac{1}{2};1;k^{2}\right)}}
\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{\Pochhammersym{-\tfrac{1}{2}}{m}\Pochhammersym{\tfrac{1}{2}}{m}}{m!\;m!}k^{2m} = \frac{\pi}{2}\genhyperF{2}{1}@{-\tfrac{1}{2},\tfrac{1}{2}}{1}{k^{2}}		 

(Pi)/(2)*sum((pochhammer(-(1)/(2), m)*pochhammer((1)/(2), m))/(factorial(m)*factorial(m))*(k)^(2*m), m = 0..infinity) = (Pi)/(2)*hypergeom([-(1)/(2),(1)/(2)], [1], (k)^(2))

Divide[Pi,2]*Sum[Divide[Pochhammer[-Divide[1,2], m]*Pochhammer[Divide[1,2], m],(m)!*(m)!]*(k)^(2*m), {m, 0, Infinity}, GenerateConditions->None] == Divide[Pi,2]*HypergeometricPFQ[{-Divide[1,2],Divide[1,2]}, {1}, (k)^(2)]
Failure Successful
Failed [2 / 3]
Result: Float(-infinity)-1.343854232*I
Test Values: {k = 2}


Result: Float(-infinity)-2.498348127*I
Test Values: {k = 3}

Successful [Tested: 3]
19.5.E3 D ⁡ ( k ) = π 4 ⁢ ∑ m = 0 ∞ ( 3 2 ) m ⁢ ( 1 2 ) m ( m + 1 ) ! ⁢ m ! ⁢ k 2 ⁢ m complete-elliptic-integral-D 𝑘 𝜋 4 superscript subscript 𝑚 0 Pochhammer 3 2 𝑚 Pochhammer 1 2 𝑚 𝑚 1 𝑚 superscript 𝑘 2 𝑚 {\displaystyle{\displaystyle D\left(k\right)=\frac{\pi}{4}\sum_{m=0}^{\infty}% \frac{{\left(\tfrac{3}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{(m+1)!\;% m!}k^{2m}}}
\compellintDk@{k} = \frac{\pi}{4}\sum_{m=0}^{\infty}\frac{\Pochhammersym{\tfrac{3}{2}}{m}\Pochhammersym{\tfrac{1}{2}}{m}}{(m+1)!\;m!}k^{2m}		 

(EllipticK(k) - EllipticE(k))/(k)^2 = (Pi)/(4)*sum((pochhammer((3)/(2), m)*pochhammer((1)/(2), m))/(factorial(m + 1)*factorial(m))*(k)^(2*m), m = 0..infinity)

Divide[EllipticK[(k)^2] - EllipticE[(k)^2], (k)^4] == Divide[Pi,4]*Sum[Divide[Pochhammer[Divide[3,2], m]*Pochhammer[Divide[1,2], m],(m + 1)!*(m)!]*(k)^(2*m), {m, 0, Infinity}, GenerateConditions->None]
Failure Failure Error
Failed [3 / 3]
Result: Indeterminate
Test Values: {Rule[k, 1]}


Result: Complex[-0.08185805455243848, 0.4541460103381727]
Test Values: {Rule[k, 2]}

... skip entries to safe data
19.5.E3 π 4 ⁢ ∑ m = 0 ∞ ( 3 2 ) m ⁢ ( 1 2 ) m ( m + 1 ) ! ⁢ m ! ⁢ k 2 ⁢ m = π 4 ⁢ F 1 2 ⁡ ( 3 2 , 1 2 ; 2 ; k 2 ) 𝜋 4 superscript subscript 𝑚 0 Pochhammer 3 2 𝑚 Pochhammer 1 2 𝑚 𝑚 1 𝑚 superscript 𝑘 2 𝑚 𝜋 4 Gauss-hypergeometric-F-as-2F1 3 2 1 2 2 superscript 𝑘 2 {\displaystyle{\displaystyle\frac{\pi}{4}\sum_{m=0}^{\infty}\frac{{\left(% \tfrac{3}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{(m+1)!\;m!}k^{2m}=% \frac{\pi}{4}{{}_{2}F_{1}}\left(\tfrac{3}{2},\tfrac{1}{2};2;k^{2}\right)}}
\frac{\pi}{4}\sum_{m=0}^{\infty}\frac{\Pochhammersym{\tfrac{3}{2}}{m}\Pochhammersym{\tfrac{1}{2}}{m}}{(m+1)!\;m!}k^{2m} = \frac{\pi}{4}\genhyperF{2}{1}@{\tfrac{3}{2},\tfrac{1}{2}}{2}{k^{2}}		 

(Pi)/(4)*sum((pochhammer((3)/(2), m)*pochhammer((1)/(2), m))/(factorial(m + 1)*factorial(m))*(k)^(2*m), m = 0..infinity) = (Pi)/(4)*hypergeom([(3)/(2),(1)/(2)], [2], (k)^(2))

Divide[Pi,4]*Sum[Divide[Pochhammer[Divide[3,2], m]*Pochhammer[Divide[1,2], m],(m + 1)!*(m)!]*(k)^(2*m), {m, 0, Infinity}, GenerateConditions->None] == Divide[Pi,4]*HypergeometricPFQ[{Divide[3,2],Divide[1,2]}, {2}, (k)^(2)]
Failure Successful
Failed [3 / 3]
Result: Float(infinity)+Float(infinity)*I
Test Values: {k = 1}


Result: Float(infinity)+.6055280139*I
Test Values: {k = 2}

... skip entries to safe data
Failed [1 / 3]
Result: Indeterminate
Test Values: {Rule[k, 1]}

19.5.E4 Π ⁡ ( α 2 , k ) = π 2 ⁢ ∑ n = 0 ∞ ( 1 2 ) n n ! ⁢ ∑ m = 0 n ( 1 2 ) m m ! ⁢ k 2 ⁢ m ⁢ α 2 ⁢ n - 2 ⁢ m complete-elliptic-integral-third-kind-Pi superscript 𝛼 2 𝑘 𝜋 2 superscript subscript 𝑛 0 Pochhammer 1 2 𝑛 𝑛 superscript subscript 𝑚 0 𝑛 Pochhammer 1 2 𝑚 𝑚 superscript 𝑘 2 𝑚 superscript 𝛼 2 𝑛 2 𝑚 {\displaystyle{\displaystyle\Pi\left(\alpha^{2},k\right)=\frac{\pi}{2}\sum_{n=% 0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{n}}}{n!}\sum_{m=0}^{n}\frac{{% \left(\tfrac{1}{2}\right)_{m}}}{m!}k^{2m}\alpha^{2n-2m}}}
\compellintPik@{\alpha^{2}}{k} = \frac{\pi}{2}\sum_{n=0}^{\infty}\frac{\Pochhammersym{\tfrac{1}{2}}{n}}{n!}\sum_{m=0}^{n}\frac{\Pochhammersym{\tfrac{1}{2}}{m}}{m!}k^{2m}\alpha^{2n-2m}		 

EllipticPi((alpha)^(2), k) = (Pi)/(2)*sum((pochhammer((1)/(2), n))/(factorial(n))*sum((pochhammer((1)/(2), m))/(factorial(m))*(k)^(2*m)* (alpha)^(2*n - 2*m), m = 0..n), n = 0..infinity)

EllipticPi[\[Alpha]^(2), (k)^2] == Divide[Pi,2]*Sum[Divide[Pochhammer[Divide[1,2], n],(n)!]*Sum[Divide[Pochhammer[Divide[1,2], m],(m)!]*(k)^(2*m)* \[Alpha]^(2*n - 2*m), {m, 0, n}, GenerateConditions->None], {n, 0, Infinity}, GenerateConditions->None]
Aborted Failure Error Skipped - Because timed out
19.5.E4 π 2 ⁢ ∑ n = 0 ∞ ( 1 2 ) n n ! ⁢ ∑ m = 0 n ( 1 2 ) m m ! ⁢ k 2 ⁢ m ⁢ α 2 ⁢ n - 2 ⁢ m = π 2 ⁢ F 1 ⁡ ( 1 2 ; 1 2 , 1 ; 1 ; k 2 , α 2 ) 𝜋 2 superscript subscript 𝑛 0 Pochhammer 1 2 𝑛 𝑛 superscript subscript 𝑚 0 𝑛 Pochhammer 1 2 𝑚 𝑚 superscript 𝑘 2 𝑚 superscript 𝛼 2 𝑛 2 𝑚 𝜋 2 Appell-F-1 1 2 1 2 1 1 superscript 𝑘 2 superscript 𝛼 2 {\displaystyle{\displaystyle\frac{\pi}{2}\sum_{n=0}^{\infty}\frac{{\left(% \tfrac{1}{2}\right)_{n}}}{n!}\sum_{m=0}^{n}\frac{{\left(\tfrac{1}{2}\right)_{m% }}}{m!}k^{2m}\alpha^{2n-2m}=\frac{\pi}{2}{F_{1}}\left(\tfrac{1}{2};\tfrac{1}{2% },1;1;k^{2},\alpha^{2}\right)}}
\frac{\pi}{2}\sum_{n=0}^{\infty}\frac{\Pochhammersym{\tfrac{1}{2}}{n}}{n!}\sum_{m=0}^{n}\frac{\Pochhammersym{\tfrac{1}{2}}{m}}{m!}k^{2m}\alpha^{2n-2m} = \frac{\pi}{2}\AppellF{1}@{\tfrac{1}{2}}{\tfrac{1}{2}}{1}{1}{k^{2}}{\alpha^{2}}		 

Error

Divide[Pi,2]*Sum[Divide[Pochhammer[Divide[1,2], n],(n)!]*Sum[Divide[Pochhammer[Divide[1,2], m],(m)!]*(k)^(2*m)* \[Alpha]^(2*n - 2*m), {m, 0, n}, GenerateConditions->None], {n, 0, Infinity}, GenerateConditions->None] == Divide[Pi,2]*AppellF[1, , Divide[1,2], Divide[1,2], 1, 1]*(k)^(2)*\[Alpha]^(2)
Missing Macro Error Failure Skip - symbolical successful subtest Skipped - Because timed out
19.5.E5 q = exp ⁡ ( - π ⁢ K ′ ⁡ ( k ) / K ⁡ ( k ) ) 𝑞 𝜋 complementary-complete-elliptic-integral-first-kind-K 𝑘 complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle q=\exp\left(-\pi{K^{\prime}}\left(k\right)/K\left% (k\right)\right)}}
q = \exp@{-\pi\ccompellintKk@{k}/\compellintKk@{k}}		 r = 1 16 ⁢ k 2 , 0 ≤ k , k ≤ 1 formulae-sequence 𝑟 1 16 superscript 𝑘 2 formulae-sequence 0 𝑘 𝑘 1 {\displaystyle{\displaystyle r=\frac{1}{16}k^{2},0\leq k,k\leq 1}} 
(exp(- Pi*EllipticCK(k)/EllipticK(k))) = exp(- Pi*EllipticCK(k)/EllipticK(k))

(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]) == Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]
Successful Successful - Successful [Tested: 1]
19.5.E7 λ = ( 1 - k ′ ) / ( 2 ⁢ ( 1 + k ′ ) ) 𝜆 1 superscript 𝑘 ′ 2 1 superscript 𝑘 ′ {\displaystyle{\displaystyle\lambda=(1-\sqrt{k^{\prime}})/(2(1+\sqrt{k^{\prime% }}))}}
\lambda = (1-\sqrt{k^{\prime}})/(2(1+\sqrt{k^{\prime}}))		 

lambda = (1 -sqrt(sqrt(1 - (k)^(2))))/(2*(1 +sqrt(sqrt(1 - (k)^(2)))))
 
\[Lambda] == (1 -Sqrt[Sqrt[1 - (k)^(2)]])/(2*(1 +Sqrt[Sqrt[1 - (k)^(2)]]))
 Skipped - no semantic math Skipped - no semantic math - -
19.5.E8 K ⁡ ( k ) = π 2 ⁢ ( 1 + 2 ⁢ ∑ n = 1 ∞ q n 2 ) 2 complete-elliptic-integral-first-kind-K 𝑘 𝜋 2 superscript 1 2 superscript subscript 𝑛 1 superscript 𝑞 superscript 𝑛 2 2 {\displaystyle{\displaystyle K\left(k\right)=\frac{\pi}{2}\left(1+2\sum_{n=1}^% {\infty}q^{n^{2}}\right)^{2}}}
\compellintKk@{k} = \frac{\pi}{2}\left(1+2\sum_{n=1}^{\infty}q^{n^{2}}\right)^{2}		 | q | < 1 𝑞 1 {\displaystyle{\displaystyle|q|<1}} 
EllipticK(k) = (Pi)/(2)*(1 + 2*sum((exp(- Pi*EllipticCK(k)/EllipticK(k)))^((n)^(2)), n = 1..infinity))^(2)

EllipticK[(k)^2] == Divide[Pi,2]*((1 + 2*Sum[(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^((n)^(2)), {n, 1, Infinity}, GenerateConditions->None]))^(2)
Failure Failure Error
Failed [1 / 3]
Result: DirectedInfinity[]
Test Values: {Rule[k, 1]}

19.5.E9 E ⁡ ( k ) = K ⁡ ( k ) + 2 ⁢ π 2 K ⁡ ( k ) ⁢ ∑ n = 1 ∞ ( - 1 ) n ⁢ n 2 ⁢ q n 2 1 + 2 ⁢ ∑ n = 1 ∞ ( - 1 ) n ⁢ q n 2 complete-elliptic-integral-second-kind-E 𝑘 complete-elliptic-integral-first-kind-K 𝑘 2 superscript 𝜋 2 complete-elliptic-integral-first-kind-K 𝑘 superscript subscript 𝑛 1 superscript 1 𝑛 superscript 𝑛 2 superscript 𝑞 superscript 𝑛 2 1 2 superscript subscript 𝑛 1 superscript 1 𝑛 superscript 𝑞 superscript 𝑛 2 {\displaystyle{\displaystyle E\left(k\right)=K\left(k\right)+\frac{2\pi^{2}}{K% \left(k\right)}\,\frac{\sum_{n=1}^{\infty}(-1)^{n}n^{2}q^{n^{2}}}{1+2\sum_{n=1% }^{\infty}(-1)^{n}q^{n^{2}}}}}
\compellintEk@{k} = \compellintKk@{k}+\frac{2\pi^{2}}{\compellintKk@{k}}\,\frac{\sum_{n=1}^{\infty}(-1)^{n}n^{2}q^{n^{2}}}{1+2\sum_{n=1}^{\infty}(-1)^{n}q^{n^{2}}}		 | q | < 1 𝑞 1 {\displaystyle{\displaystyle|q|<1}} 
EllipticE(k) = EllipticK(k)+(2*(Pi)^(2))/(EllipticK(k))*(sum((- 1)^(n)* (n)^(2)*(exp(- Pi*EllipticCK(k)/EllipticK(k)))^((n)^(2)), n = 1..infinity))/(1 + 2*sum((- 1)^(n)*(exp(- Pi*EllipticCK(k)/EllipticK(k)))^((n)^(2)), n = 1..infinity))

EllipticE[(k)^2] == EllipticK[(k)^2]+Divide[2*(Pi)^(2),EllipticK[(k)^2]]*Divide[Sum[(- 1)^(n)* (n)^(2)*(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^((n)^(2)), {n, 1, Infinity}, GenerateConditions->None],1 + 2*Sum[(- 1)^(n)*(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^((n)^(2)), {n, 1, Infinity}, GenerateConditions->None]]
Failure Failure Error Skipped - Because timed out
19.5.E10 K ⁡ ( k ) = π 2 ⁢ ∏ m = 1 ∞ ( 1 + k m ) complete-elliptic-integral-first-kind-K 𝑘 𝜋 2 superscript subscript product 𝑚 1 1 subscript 𝑘 𝑚 {\displaystyle{\displaystyle K\left(k\right)=\frac{\pi}{2}\prod_{m=1}^{\infty}% (1+k_{m})}}
\compellintKk@{k} = \frac{\pi}{2}\prod_{m=1}^{\infty}(1+k_{m})		 

EllipticK(k) = (Pi)/(2)*product(1 + k[m], m = 1..infinity)

EllipticK[(k)^2] == Divide[Pi,2]*Product[1 + Subscript[k, m], {m, 1, Infinity}, GenerateConditions->None]
Failure Failure Error
Failed [30 / 30]
Result: Plus[DirectedInfinity[], Times[-1.5707963267948966, NProduct[Plus[1, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]
Test Values: {m, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[k, 1], Rule[Subscript[k, m], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Plus[Complex[0.8428751774062981, -1.0782578237498217], Times[-1.5707963267948966, NProduct[Plus[1, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]
Test Values: {m, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[k, 2], Rule[Subscript[k, m], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.5.E11 k m + 1 = 1 - 1 - k m 2 1 + 1 - k m 2 subscript 𝑘 𝑚 1 1 1 superscript subscript 𝑘 𝑚 2 1 1 superscript subscript 𝑘 𝑚 2 {\displaystyle{\displaystyle k_{m+1}=\frac{1-\sqrt{1-k_{m}^{2}}}{1+\sqrt{1-k_{% m}^{2}}}}}
k_{m+1} = \frac{1-\sqrt{1-k_{m}^{2}}}{1+\sqrt{1-k_{m}^{2}}}		 

k[m + 1] = (1 -sqrt(1 - (k[m])^(2)))/(1 +sqrt(1 - (k[m])^(2)))
 
Subscript[k, m + 1] == Divide[1 -Sqrt[1 - (Subscript[k, m])^(2)],1 +Sqrt[1 - (Subscript[k, m])^(2)]]
 Skipped - no semantic math Skipped - no semantic math - -
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