Exercise:
Considera il quadrato unitario con vertici +ii. Determina la sua immagine sotto alle operazioni zmapsto fracz e zmapsto z^.

Solution:
Sotto z^ l'immagine è questa ``cupola'' composta da due archi di parabola e un segmento orizzontale. center tikzpicturescale. axisxcmycm axis linesmiddle enlargelimitsfalse axis line styleshorten -pt shorten -pt xlabel styleanchorwest atticklabel* cs:. xshiftpt ylabel styleanchorsouth atticklabel* cs:. yshiftpt xlabel Re ylabel Im xmin- xmax ymin- ymax ytick--... xtick--... minor x tick num minor y tick num minor grid styleblack! major grid styleblack! grid yminorgrids xminorgrids major tick lengthpt every tick/.style semithick axis scopeshift fillcolorblackfill opacity. line widthptsmoothsamplesdomain: plotx^-*x -- -- cycle; fillcolorblackfill opacity. line widthptsmoothsamplesdomain: plot-x^+*x -- -- cycle; drawcolorblack line widthptsmoothsamplesdomain: plot-x^*x; drawcolorblack line widthptsmoothsamplesdomain: plotx^-*x; drawline widthpt - -- ; scope tikzpicture center Sotto l'inversione+riflessione zmapsto fracz invece otteniamo la figura center tikzpicturescale. axisxcmycm axis linesmiddle enlargelimitsfalse axis line styleshorten -pt shorten -pt xlabel styleanchorwest atticklabel* cs:. xshiftpt ylabel styleanchorsouth atticklabel* cs:. yshiftpt xlabel Re ylabel Im xmin- xmax ymin- ymax ytick--... xtick--... minor x tick num minor y tick num minor grid styleblack! major grid styleblack! grid yminorgrids xminorgrids major tick lengthpt every tick/.style semithick axis scopeshift fillfillblackfill opacity. -- -- - -- - -- - -- - -- cycle; drawline widthpt -- ; drawline widthpt - -- -; drawline widthpt - arc -::.; drawline widthpt .-. arc -::.; fillfillblackfill opacity. .-. arc -::. -- - -- cycle; fillfillblackfill opacity. - arc -::. -- - -- cycle; scope tikzpicture center