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JacobiZeta - Jacobi's Zeta function
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Calling Sequence
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JacobiZeta(z, k)
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Parameters
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z
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algebraic expression
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k
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algebraic expression
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Description
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JacobiZeta is defined by:
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FunctionAdvisor(definition, JacobiZeta);
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![[JacobiZeta(z, k) = Diff(ln(JacobiTheta4((1/2)*Pi*z/EllipticK(k), EllipticNome(k))), z), `with no restrictions on `(z, k)]](/support/helpjp/helpview.aspx?si=6305/file00678/math55.png)
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which is essentially the logarithmic derivative of JacobiTheta4.
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JacobiZeta(z,k) is a periodic function of  with period 
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Examples
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![[JacobiZeta(-z, k) = -JacobiZeta(z, k), JacobiZeta(z, -k) = JacobiZeta(z, k), JacobiZeta(0, k) = 0, JacobiZeta(z, 0) = 0, JacobiZeta(z, 1) = tanh(z), JacobiZeta(z, infinity) = infinity+infinity*I, [JacobiZeta(2*_n1*EllipticK(k), k) = 0, _n1::integer]]](/support/helpjp/helpview.aspx?si=6305/file00678/math84.png)
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![[JacobiZeta(z, k) = Sum(-2*Pi*EllipticNome(k)^_k1*sin(_k1*Pi*z/EllipticK(k))/(EllipticK(k)*(-1+EllipticNome(k)^(2*_k1))), _k1 = 1 .. infinity), `with no restrictions on `(k, z)]](/support/helpjp/helpview.aspx?si=6305/file00678/math91.png)
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