Algebraic and Analytic Methods - 1.10 Functions of a Complex Variable

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
1.10.E20	${\displaystyle{\displaystyle|\ln\left(1+a_{n}(z)\right)|\leq M_{n}}}$ |\ln@{1+a_{n}(z)}| \leq M_{n}	${\displaystyle{\displaystyle n\geq N}}$	abs(ln(1 + a[n](z))) <= M[n]	Abs[Log[1 + Subscript[a, n][z]]] <= Subscript[M, n]	Failure	Failure	Failed [126 / 300] Result: .7588760888 <= -1.5 Test Values: {z = 1/2*3^(1/2)+1/2*I, M[n] = -1.5, a[n] = 1/2*3^(1/2)+1/2*I, n = 1} Result: .7588760888 <= -1.5 Test Values: {z = 1/2*3^(1/2)+1/2*I, M[n] = -1.5, a[n] = 1/2*3^(1/2)+1/2*I, n = 2} Result: .7588760888 <= -1.5 Test Values: {z = 1/2*3^(1/2)+1/2*I, M[n] = -1.5, a[n] = 1/2*3^(1/2)+1/2*I, n = 3} Result: 1.465287519 <= -1.5 Test Values: {z = 1/2*3^(1/2)+1/2*I, M[n] = -1.5, a[n] = -1/2+1/2*I*3^(1/2), n = 1} ... skip entries to safe data	Failed [246 / 300] Result: LessEqual[0.7588760887069661, Complex[0.8660254037844387, 0.49999999999999994]] Test Values: {Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[M, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: LessEqual[0.7588760887069661, Complex[0.8660254037844387, 0.49999999999999994]] Test Values: {Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[M, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
1.10.E21	${\displaystyle{\displaystyle\sum^{\infty}_{n=1}M_{n}<\infty}}$ \sum^{\infty}_{n=1}M_{n} < \infty		sum(M[n](<)*infinity, n = 1..infinity)	Sum[Subscript[M, n][<]*Infinity, {n, 1, Infinity}, GenerateConditions->None]	Skipped - no semantic math	Skipped - no semantic math	-	-
1.10.E22	${\displaystyle{\displaystyle P(z)=\prod^{\infty}_{n=1}\left(1-\frac{z}{z_{n}}\right)e^{z/z_{n}}}}$ P(z) = \prod^{\infty}_{n=1}\left(1-\frac{z}{z_{n}}\right)e^{z/z_{n}}		P(z) = product((1 -(z)/(z[n]))*exp(z/z[n]), n = 1..infinity)	P[z] == Product[(1 -Divide[z,Subscript[z, n]])*Exp[z/Subscript[z, n]], {n, 1, Infinity}, GenerateConditions->None]	Skipped - no semantic math	Skipped - no semantic math	-	-