Add taylorterm(), the n-th term in Taylor series

Also, correct innerproduct to the more usual form <u,v>.
Leedehai · Dec 10, 2023 · faaacd6 · faaacd6
1 parent 142fbde
commit faaacd6
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diff --git a/README.md b/README.md
@@ -1,6 +1,6 @@
 :green_book: The [manual](https://github.com/Leedehai/typst-physics/blob/master/physica-manual.pdf).
 <p align="center">
-<img width="545" alt="logo" src="https://github.com/Leedehai/typst-physics/assets/18319900/781d5747-4842-49f6-bb41-a1fd4d17a1ff">
+<img width="545" alt="logo" src="https://github.com/Leedehai/typst-physics/assets/18319900/c703f3db-1b84-4187-ab5f-3929febc89ac">
 </p>
 
 # The physica package for Typst
@@ -21,7 +21,7 @@ natural sciences, including:
 * Dirac braket notations,
 * Common math functions,
 * Differentials and derivatives, including partial derivatives of mixed orders with automatic order summation,
-* Familiar "h-bar", tensor abstract index notations, isotopes,
+* Familiar "h-bar", tensor abstract index notations, isotopes, Taylor series term,
 * Signal sequences i.e. digital timing diagrams.
 
 :warning: [Typst](https://typst.app) is in beta and evolving, and this package

diff --git a/background3.png b/background3.png
diff --git a/physica-manual.pdf b/physica-manual.pdf
diff --git a/physica-manual.typ b/physica-manual.typ
@@ -27,8 +27,8 @@
 
 #align(center)[
   Leedehai \
-  #linkurl("GitHub repo", "https://github.com/leedehai/typst-physics") |
-  #linkurl("Typst index", "https://typst.app/docs/packages/")
+  #linkurl("GitHub", "https://github.com/leedehai/typst-physics") |
+  #linkurl("Typst", "https://typst.app/docs/packages/")
 ]
 
 #set par(justify: true)
@@ -211,6 +211,14 @@ All symbols need to be used in *math mode* `$...$`.
   [`cprod`],
   [`a cprod b` #sym.arrow $a cprod b$],
   [cross product],
+
+  [`innerproduct`],
+  [`iprod`],
+  [
+    `iprod(u, v)` #sym.arrow $iprod(u, v)$ \
+    `iprod(sum_i a_i, b)` \ #sym.arrow $iprod(sum_i a_i, b)$
+  ],
+  [inner product],
 )
 
 == Matrix notations
@@ -399,22 +407,6 @@ xmat(2, 2, #g)`)
   ],
   [ketbra],
 
-  [`innerproduct(`_a_, _b_`)`],
-  [`iprod`],
-  [
-    `iprod(a), iprod(u, v)` \ #sym.arrow $iprod(a), iprod(u, v)$ \
-    `iprod(a, vec(1,2))` #sym.arrow $iprod(a, vec(1,2))$
-  ],
-  [innerproduct\ = braket],
-
-  [`outerproduct(`_a_, _b_`)`],
-  [`oprod`],
-  [
-    `oprod(a), oprod(u, v)` \ #sym.arrow $oprod(a), oprod(u, v)$ \
-    `oprod(a, vec(1,2))` #sym.arrow $oprod(a, vec(1,2))$
-  ],
-  [outerproduct\ = ketbra],
-
   [`matrixelement(`_n_, _M_, _m_`)`],
   [`mel`],
   [
@@ -852,6 +844,51 @@ $ isotope("Bi",a:211,z:83) --> isotope("Tl",a:207,z:81) + isotope("He",a:4,z:2)
 *(4)* #hl(`isotope("Tl",a:207,z:81) --> isotope("Pb",a:207,z:82) + isotope(e,a:0,z:-1)`)
 $ isotope("Tl",a:207,z:81) --> isotope("Pb",a:207,z:82) + isotope(e,a:0,z:-1) $
 
+=== The n-th term in Taylor series
+
+#v(1em)
+
+Function: `taylorterm(`_func_, _x_, _x0_, _idx_`)`.
+- _func_: the function e.g. `f`, `(f+g)`,
+- _x_: the variable name e.g. `x`,
+- _x0_: the variable value at the expansion point e.g. `x_0`, `(1+a)`,
+- _idx_: the index of the term, e.g. `0`, `1`, `2`, `n`, `(n+1)`.
+If _x0_ or _idx_ is in a parenthesis e.g. `(1+k)`, then the function
+automatically removes the outer parenthesis where appropriate.
+
+#align(center, [*Examples*])
+
+#grid(
+  columns: (50%, 50%),
+  row-gutter: 1em,
+  column-gutter: 2em,
+  [
+    *(1)* #hl(`taylorterm(f,x,x_0,0)`) \
+    $ taylorterm(f,x,x_0,0) $
+  ],
+  [
+    *(2)* #hl(`taylorterm(f,x,x_0,1)`) \
+    $ taylorterm(f,x,x_0,1) $
+  ],
+  [
+    *(3)* #hl(`taylorterm(f,x,(1+a),2)`) \
+    $ taylorterm(f,x,(1+a),2) $
+  ],
+  [
+    *(4)* #hl(`taylorterm(f,x,x_0,n)`) \
+    $ taylorterm(f,x,x_0,n) $
+  ],
+  [
+    *(5)* #hl(`taylorterm(F,x^nu,x^nu_0,n)`) \
+    $ taylorterm(F,x^nu,x^nu_0,n) $
+  ],
+  [
+    *(6)* #hl(`taylorterm(f_p,x,x_0,(n+1))`) \
+    $ taylorterm(f_p,x,x_0,(n+1)) $
+  ],
+)
+
+
 === Signal sequences (digital timing diagrams)
 
 In engineering, people often need to draw digital timing diagrams for signals, like $signals("1|0|1|0")$.

diff --git a/physica.html b/physica.html
@@ -0,0 +1,27 @@
+<html>
+<head>
+  <style>
+    .parent {
+      position: relative;
+      top: 0;
+      left: 0;
+    }
+    .image1 {
+      position: relative;
+      top: 0;
+      left: 0;
+    }
+    .image2 {
+      position: absolute;
+      top: 100px;
+      left: 195px;
+      border: 2px solid white;
+    }
+  </style>
+</head>
+<body>
+<div class="parent">
+  <img class="image1" src="./background3.png" />
+  <img class="image2" src="./physica_thin.png" />
+</div>
+</body>
diff --git a/physica.typ b/physica.typ
@@ -206,6 +206,11 @@
 #let crossproduct = $times$
 #let cprod = crossproduct
 
+#let innerproduct(u, v) = {
+  $lr(angle.l #u, #v angle.r)$
+}
+#let iprod = innerproduct
+
 // == Matrices
 
 #let matrixdet(..sink) = {
@@ -288,8 +293,8 @@
 #let jmat = jacobianmatrix
 
 #let hessianmatrix(fs, xs, delim:"(") = {
-  assert(type(fs) == array, message: "expecting a one-element array")
-  assert(fs.len() == 1, message: "expecting only one function name")
+  assert(type(fs) == array, message: "usage: hessianmatrix(f; x, y...)")
+  assert(fs.len() == 1, message: "usage: hessianmatrix(f; x, y...)")
   let f = fs.at(0)
   assert(type(xs) == array, message: "expecting an array of variable names")
   let row_arrays = ()  // array of arrays
@@ -364,11 +369,6 @@
   $ lr(bar.v ket#h(0pt)mid(angle.r#h(0pt)angle.l)#h(0pt)bra bar.v) $
 })
 
-#let innerproduct = braket
-#let iprod = innerproduct
-#let outerproduct = ketbra
-#let oprod = outerproduct
-
 #let matrixelement(n, M, m) = style(styles => {
   $ lr(angle.l #n#h(0pt)mid(bar.v)#h(0pt)#M#h(0pt)mid(bar.v)#h(0pt)#m angle.r) $
 })
@@ -663,6 +663,25 @@
   math.attach((T,hphantom(sym.zwj)).join(), t: uppers.join(), b: lowers.join())
 }
 
+#let taylorterm(fn, xv, x0, idx) = {
+  let noparen(expr) = {
+    if type(expr) == content and expr.func() == math.lr {
+      let children = expr.at("body").at("children")
+      children.slice(1, children.len() - 1).join()
+    } else {
+      expr
+    }
+  }
+
+  if idx == [0] or idx == 0 {
+    $fn (noparen(x0))$
+  } else if idx == [1] or idx == 1 {
+    $fn^((1)) (noparen(x0))(xv - x0)$
+  } else {
+    $frac(fn^((noparen(idx))) (noparen(x0)), idx !)(xv - x0)^noparen(idx)$
+  }
+}
+
 #let isotope(element, /*atomic mass*/a: none, /*atomic number*/z: none) = {
   $attach(upright(element), tl: #a, bl: #z)$
 }

diff --git a/physica_thin.png b/physica_thin.png