name Maclaurin series definition Order fivel = (1, π / 5) z 0 ÷ 0! + z 5 ÷ 5! + z 10 ÷ 10! + z 15 ÷ 15! … 1⁄5 ( l 0 exp (l 0z) + l 0 exp (l 2z) + l 0 exp (l 4z) + l 0 exp (l 6z) + l 0 exp (l 8z) ) z 0 ÷ 0! − z 5 ÷ 5! + z 10 ÷ 10! − z 15 ÷ 15! … 1⁄5 ( l 0 exp (l 1z) + l 0 exp (l 3z) + l 0 exp (l 5z) + l 0 exp (l 7z) + l 0 exp (l 9z) ) z 1 ÷ 1! + z 6 ÷ 6! + z 11 ÷ 11! + z 16 ÷ 16! … 1⁄5 ( l 0 exp (l 0z) + l 2 exp (l 8z) + l 4 exp (l 6z) + l 6 exp (l 4z) + l 8 exp (l 2z) ) z 1 ÷ 1! − z 6 ÷ 6! + z 11 ÷ 11! − z 16 ÷ 16! … 1⁄5 ( l 1 exp (l 9z) + l 3 exp (l 7z) + l 5 exp (l 5z) + l 7 exp (l 3z) + l 9 exp (l 1z) ) z 2 ÷ 2! + z 7 ÷ 7! + z 12 ÷ 12! + z 17 ÷ 17! … 1⁄5 ( l 0 exp (l 0z) + l 2 exp (l 4z) + l 4 exp (l 8z) + l 6 exp (l 2z) + l 8 exp (l 6z) ) z 2 ÷ 2! − z 7 ÷ 7! + z 12 ÷ 12! − z 17 ÷ 17! … 1⁄5 ( l 0 exp (l 5z) + l 2 exp (l 9z) + l 4 exp (l 3z) + l 6 exp (l 7z) + l 8 exp (l 1z) ) z 3 ÷ 3! + z 8 ÷ 8! + z 13 ÷ 13! + z 18 ÷ 18! … 1⁄5 ( l 0 exp (l 0z) + l 2 exp (l 6z) + l 4 exp (l 2z) + l 6 exp (l 8z) + l 8 exp (l 4z) ) z 3 ÷ 3! − z 8 ÷ 8! + z 13 ÷ 13! − z 18 ÷ 18! … 1⁄5 ( l 1 exp (l 3z) + l 3 exp (l 9z) + l 5 exp (l 5z) + l 7 exp (l 1z) + l 9 exp (l 7z) ) z 4 ÷ 4! + z 9 ÷ 9! + z 14 ÷ 14! + z 19 ÷ 19! … 1⁄5 ( l 0 exp (l 0z) + l 2 exp (l 2z) + l 4 exp (l 4z) + l 6 exp (l 6z) + l 8 exp (l 8z) ) z 4 ÷ 4! − z 9 ÷ 9! + z 14 ÷ 14! − z 19 ÷ 19! … 1⁄5 ( l 0 exp (l 5z) + l 2 exp (l 7z) + l 4 exp (l 9z) + l 6 exp (l 1z) + l 8 exp (l 3z) )

name Maclaurin series definition Order sixm = (1, π / 6) z 0 ÷ 0! + z 6 ÷ 6! + z 12 ÷ 12! + z 18 ÷ 18! … 1⁄6 ( m 0 exp (m 0z) + m 0 exp (m 2z) + m 0 exp (m 4z) + m 0 exp (m 6z) + m 0 exp (m 8z) + m 0 exp (m 10z) ) z 0 ÷ 0! − z 6 ÷ 6! + z 12 ÷ 12! − z 18 ÷ 18! … 1⁄6 ( m 0 exp (m 1z) + m 0 exp (m 3z) + m 0 exp (m 5z) + m 0 exp (m 7z) + m 0 exp (m 9z) + m 0 exp (m 11z) ) z 1 ÷ 1! + z 7 ÷ 7! + z 13 ÷ 13! + z 19 ÷ 19! … 1⁄6 ( m 0 exp (m 0z) + m 2 exp (m 10z) + m 4 exp (m 8z) + m 6 exp (m 6z) + m 8 exp (m 4z) + m 10 exp (m 2z) ) z 1 ÷ 1! − z 7 ÷ 7! + z 13 ÷ 13! − z 19 ÷ 19! … 1⁄6 ( m 1 exp (m 11z) + m 3 exp (m 9z) + m 5 exp (m 7z) + m 7 exp (m 5z) + m 9 exp (m 3z) + m 11 exp (m 1z) ) z 2 ÷ 2! + z 8 ÷ 8! + z 14 ÷ 14! + z 20 ÷ 20! … 1⁄6 ( m 0 exp (m 0z) + m 0 exp (m 6z) + m 4 exp (m 4z) + m 4 exp (m 10z) + m 8 exp (m 8z) + m 8 exp (m 2z) ) z 2 ÷ 2! − z 8 ÷ 8! + z 14 ÷ 14! − z 20 ÷ 20! … 1⁄6 ( m 2 exp (m 5z) + m 2 exp (m 11z) + m 6 exp (m 3z) + m 6 exp (m 9z) + m 10 exp (m 1z) + m 10 exp (m 7z) ) z 3 ÷ 3! + z 9 ÷ 9! + z 15 ÷ 15! + z 21 ÷ 21! … 1⁄6 ( m 0 exp (m 0z) + m 0 exp (m 4z) + m 0 exp (m 8z) + m 6 exp (m 2z) + m 6 exp (m 6z) + m 6 exp (m 10z) ) z 3 ÷ 3! − z 9 ÷ 9! + z 15 ÷ 15! − z 21 ÷ 21! … 1⁄6 ( m 3 exp (m 3z) + m 3 exp (m 7z) + m 3 exp (m 11z) + m 9 exp (m 1z) + m 9 exp (m 5z) + m 9 exp (m 9z) ) z 4 ÷ 4! + z 10 ÷ 10! + z 16 ÷ 16! + z 22 ÷ 22! … 1⁄6 ( m 0 exp (m 0z) + m 0 exp (m 6z) + m 4 exp (m 2z) + m 4 exp (m 8z) + m 8 exp (m 4z) + m 8 exp (m 10z) ) z 4 ÷ 4! − z 10 ÷ 10! + z 16 ÷ 16! − z 22 ÷ 22! … 1⁄6 ( m 0 exp (m 3z) + m 0 exp (m 9z) + m 4 exp (m 5z) + m 4 exp (m 11z) + m 8 exp (m 1z) + m 8 exp (m 7z) ) z 5 ÷ 5! + z 11 ÷ 11! + z 17 ÷ 17! + z 23 ÷ 23! … 1⁄6 ( m 0 exp (m 0z) + m 2 exp (m 2z) + m 4 exp (m 4z) + m 6 exp (m 6z) + m 8 exp (m 8z) + m 10 exp (m 10z) ) z 5 ÷ 5! − z 11 ÷ 11! + z 17 ÷ 17! − z 23 ÷ 23! … 1⁄6 ( m 1 exp (m 7z) + m 3 exp (m 9z) + m 5 exp (m 11z) + m 7 exp (m 1z) + m 9 exp (m 3z) + m 11 exp (m 5z) )