diff --git a/notebooks/Quadrature.ipynb b/notebooks/Quadrature.ipynb
index baa259e..ea59919 100644
--- a/notebooks/Quadrature.ipynb
+++ b/notebooks/Quadrature.ipynb
@@ -83,10 +83,10 @@
"execution_count": 1,
"metadata": {
"execution": {
- "iopub.execute_input": "2022-03-21T10:44:22.273312Z",
- "iopub.status.busy": "2022-03-21T10:44:21.898232Z",
- "iopub.status.idle": "2022-03-21T10:44:36.719355Z",
- "shell.execute_reply": "2022-03-21T10:44:36.718891Z"
+ "iopub.execute_input": "2022-04-06T18:58:20.378530Z",
+ "iopub.status.busy": "2022-04-06T18:58:19.863432Z",
+ "iopub.status.idle": "2022-04-06T18:58:37.594817Z",
+ "shell.execute_reply": "2022-04-06T18:58:37.594251Z"
}
},
"outputs": [
@@ -96,93 +96,93 @@
"\n",
"\n"
+ "\n",
+ "\n"
]
},
"execution_count": 1,
@@ -499,13 +499,13 @@
"\n",
"Suppose $x_1, …,x_j$ are the roots where $q_n(x)$ changes sign, that is,\n",
"$$\n",
- "q_n(x) = c_j (x-x_j) + O((x-x_j)^2)\n",
+ "q_n(x) = c_k (x-x_k)^{2p+1} + O((x-x_k)^{2p+2})\n",
"$$\n",
- "for $c_j ≠ 0$. Then\n",
+ "for $c_k ≠ 0$ and $k = 1,…,j$ and $p ∈ ℤ$, as$x → x_k$. Then\n",
"$$\n",
"q_n(x) (x-x_1) ⋯(x-x_j)\n",
"$$\n",
- "does not change sign.\n",
+ "does not change signs: it behaves like $c_k (x-x_k)^{2p+2} + O(x-x_k)^{2p+3}$ as $x → x_k$.\n",
"In other words:\n",
"$$\n",
"⟨q_n,(x-x_1) ⋯(x-x_j) ⟩ = \\int_a^b q_n(x) (x-x_1) ⋯(x-x_j) w(x) {\\rm d} x ≠ 0.\n",
@@ -888,7 +888,7 @@
"file_extension": ".jl",
"mimetype": "application/julia",
"name": "julia",
- "version": "1.7.1"
+ "version": "1.7.0"
}
},
"nbformat": 4,
diff --git a/src/Quadrature.jmd b/src/Quadrature.jmd
index 17c16c6..8b26c2a 100644
--- a/src/Quadrature.jmd
+++ b/src/Quadrature.jmd
@@ -151,13 +151,13 @@ first need to guarantee that the roots are distinct.
Suppose $x_1, …,x_j$ are the roots where $q_n(x)$ changes sign, that is,
$$
-q_n(x) = c_j (x-x_j) + O((x-x_j)^2)
+q_n(x) = c_k (x-x_k)^{2p+1} + O((x-x_k)^{2p+2})
$$
-for $c_j ≠ 0$. Then
+for $c_k ≠ 0$ and $k = 1,…,j$ and $p ∈ ℤ$, as$x → x_k$. Then
$$
q_n(x) (x-x_1) ⋯(x-x_j)
$$
-does not change sign.
+does not change signs: it behaves like $c_k (x-x_k)^{2p+2} + O(x-x_k)^{2p+3}$ as $x → x_k$.
In other words:
$$
⟨q_n,(x-x_1) ⋯(x-x_j) ⟩ = \int_a^b q_n(x) (x-x_1) ⋯(x-x_j) w(x) {\rm d} x ≠ 0.