Merge fix

ma5/uppg/main.fdb_latexmk

@@ -1,5 +1,5 @@
 # Fdb version 3
-["pdflatex"] 1648483010 "main.tex" "main.pdf" "main" 1648483011
+["pdflatex"] 1648483023 "main.tex" "main.pdf" "main" 1648483023
   "/usr/share/texmf-dist/fonts/map/fontname/texfonts.map" 1647844622 3524 cb3e574dea2d1052e39280babc910dc8 ""
   "/usr/share/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1647844622 1004 54797486969f23fa377b128694d548df ""
   "/usr/share/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm" 1647844622 916 f87d7c45f9c908e672703b83b72241a3 ""
@@ -86,9 +86,9 @@
   "/usr/share/texmf-dist/web2c/texmf.cnf" 1647844622 39911 2da6c67557ec033436fe5418a70a8a61 ""
   "/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map" 1648036410 4278762 af3ac1370db81a450c1118eabf1fe718 ""
   "/var/lib/texmf/web2c/pdftex/pdflatex.fmt" 1648036393 2924504 371aaf66e6dc336df8f5856c9444aca8 ""
-  "main.aux" 1648483011 2317 eb7ea0ad69ba39707b99e3ab67505cde "pdflatex"
+  "main.aux" 1648483023 2317 eb7ea0ad69ba39707b99e3ab67505cde "pdflatex"
-  "main.out" 1648483011 1159 2635baec54a16692a01d1db6cb342d5e "pdflatex"
+  "main.out" 1648483023 1159 2635baec54a16692a01d1db6cb342d5e "pdflatex"
-  "main.tex" 1648483010 13932 ba98d7bd049011d605ce0643529cdf46 ""
+  "main.tex" 1648483023 15004 9997b9302ea145b623cf22c345a25bd5 ""
   (generated)
   "main.aux"
   "main.log"

ma5/uppg/main.log

@@ -1,4 +1,4 @@
-This is pdfTeX, Version 3.141592653-2.6-1.40.23 (TeX Live 2021/Arch Linux) (preloaded format=pdflatex 2022.3.23)  28 MAR 2022 17:56
+This is pdfTeX, Version 3.141592653-2.6-1.40.23 (TeX Live 2021/Arch Linux) (preloaded format=pdflatex 2022.3.23)  28 MAR 2022 17:57
 entering extended mode
  restricted \write18 enabled.
  file:line:error style messages enabled.
@@ -391,7 +391,7 @@ Here is how much of TeX's memory you used:
  1141 hyphenation exceptions out of 8191
  60i,7n,63p,809b,330s stack positions out of 5000i,500n,10000p,200000b,80000s
 </usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx12.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmbxti10.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmr12.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmr5.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt10.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm10.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm5.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm7.pfb>
-Output written on main.pdf (7 pages, 209097 bytes).
+Output written on main.pdf (7 pages, 210659 bytes).
 PDF statistics:
  192 PDF objects out of 1000 (max. 8388607)
  145 compressed objects within 2 object streams

ma5/uppg/main.tex

@@ -384,4 +384,25 @@ $$
 = A^2 \int_0^L |\psi_1(x)z_1 + \psi_2(x)z_2|^2 dx\ = 1.0 \quad | \quad z_i \in \mathbb{C}
 $$
 
+Eftersom integranden är magnituden av ett komplext tal ges det att man kan se den som en vektor av $\mathbb{R}^2$ vilket i sin tur menas att magnituden är dess längd. Vi kan därmed skriva om magnituden med hjälp av \emph{pythagoras sats} (väljer också att skriva bort "$(x)$" från vågfunktionerna):
+
+$$
+|\psi_1(x)z_1 + \psi_2(x)z_2| = \sqrt{ \left( \psi_1 \emph{Re}(z_1) + \psi_2 \emph{Re}(z_2) \right)^2 + \left( \psi_1 \emph{Im}(z_1) + \psi_2 \emph{Im}(z_2) \right)^2 }
+$$
+
+Eftersom denna magnituden är i kvadrat får vi:
+$$
+|\psi_1(x)z_1 + \psi_2(x)z_2|^2 = \left( \psi_1 \emph{Re}(z_1) + \psi_2 \emph{Re}(z_2) \right)^2 + \left( \psi_1 \emph{Im}(z_1) + \psi_2 \emph{Im}(z_2) \right)^2 
+$$
+$$
+= \left( \psi_1 \emph{Re}(z_1) + \psi_2 \emph{Re}(z_2) \right)^2 + \left( \psi_1 \emph{Im}(z_1) + \psi_2 \emph{Im}(z_2) \right)^2 
+$$
+$$
+= ( \psi_1^2 \emph{Re}^2(z_1) + 2\psi_1 \psi_2 \emph{Re}(z_1)\emph{Re}(z_2) + \psi_2^2 \emph{Re}^2(z_2) )^2 
+$$
+$$
++ ( \psi_1^2 \emph{Im}^2(z_1) + 2\psi_1 \psi_2 \emph{Im}(z_1)\emph{Im}(z_2) + \psi_2^2 \emph{Im}^2(z_2) )^2 
+$$
+
+
 \end{document}