diff --git a/docs/report/029.ipynb b/docs/report/029.ipynb index e92896e1..7e24fbf8 100644 --- a/docs/report/029.ipynb +++ b/docs/report/029.ipynb @@ -103,10 +103,11 @@ "source": [ "As of AmpForm [v0.15](https://github.com/ComPWA/ampform/releases/tag/0.15.1), the implementation of [`BlattWeisskopfSquared`](https://ampform.readthedocs.io/0.15.x/api/ampform.dynamics/#ampform.dynamics.BlattWeisskopfSquared) contains hard-coded polynomials, see implementation [here](https://github.com/ComPWA/ampform/blob/0.15.1/src/ampform/dynamics/__init__.py#L66-L134).\n", "The motivation for this can be found in the citations mentioned in [its API documentation](https://ampform.readthedocs.io/0.15.x/api/ampform.dynamics/#ampform.dynamics.BlattWeisskopfSquared).\n", - "However, as noted by [@mmikhasenko](https://github.com/mmikhasenko) in [ComPWA/ampform#417](https://github.com/ComPWA/ampform/issues/417), the polynomials can be derived from Hankel functions of the first kind.\n", - "Von Hippel and Quigg[^1] derived a generalization of the centrifugal barrier factor $F_L$ that was introduced by {cite}`Blatt:1952ije`, showing that\n", + "However, as noted by [@mmikhasenko](https://github.com/mmikhasenko) in [ComPWA/ampform#417](https://github.com/ComPWA/ampform/issues/417), the polynomials can be derived from the spherical[^1] Hankel functions of the first kind.\n", + "Von Hippel and Quigg[^2] derived a generalization of the centrifugal barrier factor $F_L$, also called form factor, that was introduced by {cite}`Blatt:1952ije`, showing that\n", "\n", - "[^1]: See {cite}`VonHippel:1972fg`, pp. 626 and 637, and a review by COMPASS, {cite}`Ketzer:2019wmd`, p. 31." + "[^1]: See [this page](https://mathworld.wolfram.com/SphericalHankelFunctionoftheFirstKind.html) on Wolfram MathWorld for an explanation about the difference between $h_\\ell^{(1)}$ and $H_\\ell^{(1)}$.\n", + "[^2]: See {cite}`VonHippel:1972fg`, pp. 626 and 637, and a review by COMPASS, {cite}`Ketzer:2019wmd`, p. 31." ] }, { @@ -168,10 +169,10 @@ }, "source": [ "In the following, we call $F_\\ell(z)$ the _unnormalized_ Blatt–Weisskopf form factor.\n", - "Following Chung and other resources (see e.g. {cite}`Chung:1995dx`, p. 415), AmpForm implements a unitless, _normalized_ Blatt–Weisskopf factor $B_L$, meaning that $B_L(1)=1$.[^2]\n", + "Following Chung and other resources (see e.g. {cite}`Chung:1995dx`, p. 415), AmpForm implements a unitless, _normalized_ Blatt–Weisskopf factor $B_L$, meaning that $B_L(1)=1$.[^3]\n", "It can be defined in terms of $F_L$ as\n", "\n", - "[^2]: We switch to notating angular momentum with $L$ instead of $\\ell$ here to indicate that we are talking about a normalized function here." + "[^3]: We switch to notating angular momentum with $L$ instead of $\\ell$ here to indicate that we are talking about a normalized function here." ] }, { @@ -254,7 +255,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "As of writing, however, the class does not offer sophisticated algebraic simplifications for specific values or assumptions of $\\ell$ and $z$." + "This function is the general[^1] Hankel function $H_\\ell$ and the class does not offer algebraic simplifications for specific values or assumptions of $\\ell$ and $z$." ] }, { @@ -288,7 +289,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "To implement Equation {eq}`hankel-sum`, we have to define a specific [`@unevaluated`](https://ampform.readthedocs.io/0.15.x/api/ampform.sympy/#ampform.sympy.unevaluated) expression class. The following class evaluates to the sum given in Equation {eq}`hankel-sum`.\n", + "To implement Equation {eq}`hankel-sum` for the _spherical_ Hankel function, we have to define a specific [`@unevaluated`](https://ampform.readthedocs.io/0.15.x/api/ampform.sympy/#ampform.sympy.unevaluated) expression class. The following class evaluates to the sum given in Equation {eq}`hankel-sum`.\n", "The [`doit()`](https://docs.sympy.org/latest/modules/core.html#sympy.core.basic.Basic.doit) method is overwritten explicitly in order to prevent SymPy from unfolding the [`sympy.Sum`](https://docs.sympy.org/latest/modules/concrete.html#sympy.concrete.summations.Sum) on symbolic input for $\\ell$." ] },