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  "body": "\n## Summary\n\nFor a lattice $\\Lambda$ in $\\C$, the Weierstrass sigma function $\\sigma$ is an [[entire function|0G4F0XB9X2]] with a simple zero at each lattice point.\nThe Weierstrass zeta function is its logarithmic derivative $\\zeta = \\sigma'/\\sigma$.\nThey are companions of the [[Weierstrass elliptic function|361NAK68QK]] $\\wp$, since $\\zeta' = -\\wp$.\nNeither is elliptic: shifting by a period adds a constant to $\\zeta$ and multiplies $\\sigma$ by an exponential factor.\nThey are the building blocks for writing any elliptic function from its zeros and poles.\n\n## Explanations\n\n### L1 human\n\nA polynomial can be written as a product over its roots.\nThe sigma function plays this role for the lattice: it is a product over all lattice points, with exponential correction factors so that the product converges.\nTaking its logarithmic derivative gives the zeta function, a sum of simple poles, one at each lattice point.\nDifferentiating once more and changing sign gives $\\wp$, which has double poles and is truly periodic.\nSo $\\sigma$, $\\zeta$, $\\wp$ form a chain, and only the last one is elliptic.\n",
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