ex9: start, finish, and everything in between

2025-10-22 16:25:33 +02:00
parent 3d270fc245
commit 61fdb082fb
7 changed files with 694 additions and 0 deletions
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+#import "@preview/physica:0.9.6": *
+
+= problem 2
+
+== a)
+
+$h(x) := f(x) g(x)$ is odd for odd $f$ and even $g$, since
+$
+  h(x) = -f(-x) g(-x) = - (f(-x) g(-x)) = -h(-x)
+$
+
+== b)
+
+if $f$ and $g$ are both odd or even, then $h(x) := f(x) g(x)$ is even
+
+1. for the first case: $f$ and $g$ are both even
+  $
+    h(x) = f(x) g(x) = f(-x) g(-x) = h(-x)
+  $
+
+2. for the second case: $f$ and $g$ are both odd
+  $
+    h(x) & = f(x) g(x) \
+         & = (-f(-x)) (-g(-x)) = f(-x) g(-x) = h(-x)
+  $
+
+== c)
+
+if $f$ is odd and $g$ is even then both $f compose g$ and $g compose f$ are
+even, since
+1. $f(g(x)) = f(g(-x))$, since g is even
+2. $g(f(x)) = g(-f(-x)) = g(f(-x))$
+
+== d)
+
+if $f$ is odd and $L > 0$,
+$
+  integral_(-L)^L f(x) dd(x) & = integral_(-L)^0 f(x) dd(x)
+                               + integral_0^L f(x) dd(x) \
+                             & = integral_0^L f(-x) (-1) dd(x)
+                               + integral_0^L f(x) dd(x) \
+                             & = -integral_0^L f(-x) dd(x)
+                               + integral_0^L f(x) dd(x) \
+                             & = -integral_0^L f(x) dd(x)
+                               + integral_0^L f(x) dd(x) \
+                             & = 0
+$
+
+== e)
+
+if $f$ is even and $L > 0$,
+$
+  integral_(-L)^L f(x) dd(x) & = integral_(-L)^0 f(x) dd(x)
+                               + integral_0^L f(x) dd(x) \
+                             & = integral_0^L f(-x) dd(x)
+                               + integral_0^L f(x) dd(x) \
+                             & = integral_0^L f(x) dd(x)
+                               + integral_0^L f(x) dd(x) \
+                             & = 2 integral_0^L f(x) dd(x)
+$
+
+