Lamé Functions - 29.7 Asymptotic Expansions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
29.7.E3 τ 0 = 1 2 3 ⁢ ( 1 + k 2 ) ⁢ ( 1 + p 2 ) subscript 𝜏 0 1 superscript 2 3 1 superscript 𝑘 2 1 superscript 𝑝 2 {\displaystyle{\displaystyle\tau_{0}=\frac{1}{2^{3}}(1+k^{2})(1+p^{2})}}
\tau_{0} = \frac{1}{2^{3}}(1+k^{2})(1+p^{2})		 

tau[0] = (1)/((2)^(3))*(1 + (k)^(2))*(1 + (p)^(2))
 
Subscript[\[Tau], 0] == Divide[1,(2)^(3)]*(1 + (k)^(2))*(1 + (p)^(2))
 Skipped - no semantic math Skipped - no semantic math - -
29.7.E4 τ 1 = p 2 6 ⁢ ( ( 1 + k 2 ) 2 ⁢ ( p 2 + 3 ) - 4 ⁢ k 2 ⁢ ( p 2 + 5 ) ) subscript 𝜏 1 𝑝 superscript 2 6 superscript 1 superscript 𝑘 2 2 superscript 𝑝 2 3 4 superscript 𝑘 2 superscript 𝑝 2 5 {\displaystyle{\displaystyle\tau_{1}=\frac{p}{2^{6}}((1+k^{2})^{2}(p^{2}+3)-4k% ^{2}(p^{2}+5))}}
\tau_{1} = \frac{p}{2^{6}}((1+k^{2})^{2}(p^{2}+3)-4k^{2}(p^{2}+5))		 

tau[1] = (p)/((2)^(6))*((1 + (k)^(2))^(2)*((p)^(2)+ 3)- 4*(k)^(2)*((p)^(2)+ 5))
 
Subscript[\[Tau], 1] == Divide[p,(2)^(6)]*((1 + (k)^(2))^(2)*((p)^(2)+ 3)- 4*(k)^(2)*((p)^(2)+ 5))
 Skipped - no semantic math Skipped - no semantic math - -
29.7.E6 τ 2 = 1 2 10 ⁢ ( 1 + k 2 ) ⁢ ( 1 - k 2 ) 2 ⁢ ( 5 ⁢ p 4 + 34 ⁢ p 2 + 9 ) subscript 𝜏 2 1 superscript 2 10 1 superscript 𝑘 2 superscript 1 superscript 𝑘 2 2 5 superscript 𝑝 4 34 superscript 𝑝 2 9 {\displaystyle{\displaystyle\tau_{2}=\frac{1}{2^{10}}(1+k^{2})(1-k^{2})^{2}(5p% ^{4}+34p^{2}+9)}}
\tau_{2} = \frac{1}{2^{10}}(1+k^{2})(1-k^{2})^{2}(5p^{4}+34p^{2}+9)		 

tau[2] = (1)/((2)^(10))*(1 + (k)^(2))*(1 - (k)^(2))^(2)*(5*(p)^(4)+ 34*(p)^(2)+ 9)
 
Subscript[\[Tau], 2] == Divide[1,(2)^(10)]*(1 + (k)^(2))*(1 - (k)^(2))^(2)*(5*(p)^(4)+ 34*(p)^(2)+ 9)
 Skipped - no semantic math Skipped - no semantic math - -
29.7.E7 τ 3 = p 2 14 ⁢ ( ( 1 + k 2 ) 4 ⁢ ( 33 ⁢ p 4 + 410 ⁢ p 2 + 405 ) - 24 ⁢ k 2 ⁢ ( 1 + k 2 ) 2 ⁢ ( 7 ⁢ p 4 + 90 ⁢ p 2 + 95 ) + 16 ⁢ k 4 ⁢ ( 9 ⁢ p 4 + 130 ⁢ p 2 + 173 ) ) subscript 𝜏 3 𝑝 superscript 2 14 superscript 1 superscript 𝑘 2 4 33 superscript 𝑝 4 410 superscript 𝑝 2 405 24 superscript 𝑘 2 superscript 1 superscript 𝑘 2 2 7 superscript 𝑝 4 90 superscript 𝑝 2 95 16 superscript 𝑘 4 9 superscript 𝑝 4 130 superscript 𝑝 2 173 {\displaystyle{\displaystyle\tau_{3}=\frac{p}{2^{14}}((1+k^{2})^{4}(33p^{4}+41% 0p^{2}+405)-24k^{2}(1+k^{2})^{2}(7p^{4}+90p^{2}+95)+16k^{4}(9p^{4}+130p^{2}+17% 3))}}
\tau_{3} = \frac{p}{2^{14}}((1+k^{2})^{4}(33p^{4}+410p^{2}+405)-24k^{2}(1+k^{2})^{2}(7p^{4}+90p^{2}+95)+16k^{4}(9p^{4}+130p^{2}+173))		 

tau[3] = (p)/((2)^(14))*((1 + (k)^(2))^(4)*(33*(p)^(4)+ 410*(p)^(2)+ 405)- 24*(k)^(2)*(1 + (k)^(2))^(2)*(7*(p)^(4)+ 90*(p)^(2)+ 95)+ 16*(k)^(4)*(9*(p)^(4)+ 130*(p)^(2)+ 173))
 
Subscript[\[Tau], 3] == Divide[p,(2)^(14)]*((1 + (k)^(2))^(4)*(33*(p)^(4)+ 410*(p)^(2)+ 405)- 24*(k)^(2)*(1 + (k)^(2))^(2)*(7*(p)^(4)+ 90*(p)^(2)+ 95)+ 16*(k)^(4)*(9*(p)^(4)+ 130*(p)^(2)+ 173))
 Skipped - no semantic math Skipped - no semantic math - -
29.7.E8 τ 4 = 1 2 16 ⁢ ( ( 1 + k 2 ) 5 ⁢ ( 63 ⁢ p 6 + 1260 ⁢ p 4 + 2943 ⁢ p 2 + 486 ) - 8 ⁢ k 2 ⁢ ( 1 + k 2 ) 3 ⁢ ( 49 ⁢ p 6 + 1010 ⁢ p 4 + 2493 ⁢ p 2 + 432 ) + 16 ⁢ k 4 ⁢ ( 1 + k 2 ) ⁢ ( 35 ⁢ p 6 + 760 ⁢ p 4 + 2043 ⁢ p 2 + 378 ) ) subscript 𝜏 4 1 superscript 2 16 superscript 1 superscript 𝑘 2 5 63 superscript 𝑝 6 1260 superscript 𝑝 4 2943 superscript 𝑝 2 486 8 superscript 𝑘 2 superscript 1 superscript 𝑘 2 3 49 superscript 𝑝 6 1010 superscript 𝑝 4 2493 superscript 𝑝 2 432 16 superscript 𝑘 4 1 superscript 𝑘 2 35 superscript 𝑝 6 760 superscript 𝑝 4 2043 superscript 𝑝 2 378 {\displaystyle{\displaystyle\tau_{4}=\frac{1}{2^{16}}((1+k^{2})^{5}(63p^{6}+12% 60p^{4}+2943p^{2}+486)-8k^{2}(1+k^{2})^{3}(49p^{6}+1010p^{4}+2493p^{2}+432)+16% k^{4}(1+k^{2})(35p^{6}+760p^{4}+2043p^{2}+378))}}
\tau_{4} = \frac{1}{2^{16}}((1+k^{2})^{5}(63p^{6}+1260p^{4}+2943p^{2}+486)-8k^{2}(1+k^{2})^{3}(49p^{6}+1010p^{4}+2493p^{2}+432)+16k^{4}(1+k^{2})(35p^{6}+760p^{4}+2043p^{2}+378))		 

tau[4] = (1)/((2)^(16))*((1 + (k)^(2))^(5)*(63*(p)^(6)+ 1260*(p)^(4)+ 2943*(p)^(2)+ 486)- 8*(k)^(2)*(1 + (k)^(2))^(3)*(49*(p)^(6)+ 1010*(p)^(4)+ 2493*(p)^(2)+ 432)+ 16*(k)^(4)*(1 + (k)^(2))*(35*(p)^(6)+ 760*(p)^(4)+ 2043*(p)^(2)+ 378))
 
Subscript[\[Tau], 4] == Divide[1,(2)^(16)]*((1 + (k)^(2))^(5)*(63*(p)^(6)+ 1260*(p)^(4)+ 2943*(p)^(2)+ 486)- 8*(k)^(2)*(1 + (k)^(2))^(3)*(49*(p)^(6)+ 1010*(p)^(4)+ 2493*(p)^(2)+ 432)+ 16*(k)^(4)*(1 + (k)^(2))*(35*(p)^(6)+ 760*(p)^(4)+ 2043*(p)^(2)+ 378))
 Skipped - no semantic math Skipped - no semantic math - -
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