complex algebraWe don't really need these commands for our course. This is just extra fluff for those who are curious.Maple has some obvious commands for dealing with complex number algebra ("Re" and "Im" are the funny symbols, found in the Common Symbols palette, "evalc" means evaluate to a complex number, i.e., explIcitly to its real and imaginary parts). For Real and Imaginary parts use function notation: Re(3+2I), Im(3+2I)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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEjUmVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEiWkYnL0YwUSV0cnVlRicvRjNRJ2l0YWxpY0YnRjJGMi1JI21vR0YkNi1RIjtGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRidGMg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEjSW1GJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEiWkYnL0YwUSV0cnVlRicvRjNRJ2l0YWxpY0YnRjJGMi1JI21vR0YkNi1RIjtGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRidGMg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSZldmFsY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEiWkYnRjRGN0Y+Rj5GPkYrRj4=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY0LUkjbWlHRiQ2KVEmZXZhbGNGJy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUrZXhlY3V0YWJsZUdGNC8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYxUSIoRidGL0YyRjhGOy9GPlEnbm9ybWFsRicvJSZmZW5jZUdGNy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjE2NjY2NjdlbUYnLyUncnNwYWNlR0ZWLUZBNjFRIn5GJ0YvRjJGOEY7RkQvRkdGNEZIL0ZLRjRGTEZORlBGUi9GVVEmMC4wZW1GJy9GWEZpbi1GLDYpUSRleHBGJ0YvRjIvRjZGNEY4RjtGREZALUkjbW5HRiQ2KFEiM0YnRi9GMkY4RjtGRC1GQTYxUSIrRidGL0YyRjhGO0ZERmZuRkhGZ25GTEZORlBGUi9GVVEsMC4yMjIyMjIyZW1GJy9GWEZnby1GYG82KFEiMkYnRi9GMkY4RjtGRC1GQTYxUSIqRidGL0YyRjhGO0ZERmZuRkhGZ25GTEZORlBGUkZURlctRiw2KVEiSUYnRi9GMkY1RjhGO0Y9LUZBNjFRIilGJ0YvRjJGOEY7RkRGRkZIRkpGTEZORlBGUkZURldGWUZicEYvRjJGOEY7RkQ=[Right-clicking on the output of a complex number allows you to select "Complex Maps" and then Real and Imaginary parts or Argument and Modulus, which are the angle (between LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LlEifkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRUYvRjI=-\317\200 and \317\200) and radius of the polar form of a complex number, or 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 form, which applies evalc.
You can also choose Conversions, Polar to get the amplitude and principal 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Since Maple does not know if the variable LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is real or not, the Real and Imaginary part commands do not simplify in the next complex expression, but the command evalc assumes that the variables in an expression are real and conveniently evaluates the expression to its real and imaginary parts as if they were:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiYtRiw2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjpRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZELyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlMtSSVtc3VwR0YkNiUtRj02LVEvJkV4cG9uZW50aWFsRTtGJ0ZARkJGRUZHRklGS0ZNRk8vRlJRJjAuMGVtRicvRlVRLDAuMTExMTExMWVtRictRiM2Ji1JKG1mZW5jZWRHRiQ2JC1GIzYnLUkjbW5HRiQ2JFEiM0YnRkAtRj02LVEiK0YnRkBGQkZFRkdGSUZLRk1GTy9GUlEsMC4yMjIyMjIyZW1GJy9GVUZpby1GIzYmLUZibzYkUSIyRidGQC1GPTYtUTEmSW52aXNpYmxlVGltZXM7RidGQEZCRkVGR0ZJRktGTUZPRmZuL0ZVRmduLUYsNiVRIklGJ0Y2RjlGQEYrRkBGQEZgcC1GLDYlUSJ4RidGNkY5RkAvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRkBGK0ZALUY9Ni1RIjtGJ0ZARkIvRkZGOEZHRklGS0ZNRk9GZm5GVEZALUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEjUmVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEiRkYnL0YwUSV0cnVlRicvRjNRJ2l0YWxpY0YnRjJGMi1JI21vR0YkNi1RIjtGJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdGVS8lJmRlcHRoR0Zobi8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GLDYlUSNJbUYnRi9GMkY1RkFGMg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSZldmFsY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRIkZGJ0Y0RjcvRjhRJ25vcm1hbEYnRkJGQkYrRitGK0ZCMaple automatically simplifies complex numbers to their real plus imaginary part 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But Maple stops this simplification when unknown variable names appear, since they could be complex and the result would change: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while evalc will assume LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn is a real variable and go ahead and do it:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEmZXZhbGNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSIlRidGL0YyL0YzUSdub3JtYWxGJ0Y9LUkjbW9HRiQ2LVEiO0YnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRictRiw2JVEpc2ltcGxpZnlGJ0YvRjJGNUY9factoring (for 5.3)Because we like real expressions, factor does not factor quadratics with complex roots, but solve finds all the roots: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For this next fourth order characteristic equation (its roots are the four fourth roots of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JS1JI21vR0YkNi1RKiZ1bWludXMwO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZJLUkjbW5HRiQ2JFEiNEYnRjVGNUYrRjU=):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[This next input is a Maple trick, don't try this yourself---the map command is worth getting to know for serious Maple users, which by definition MAT2705 students are not:]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 real and imaginary parts of the corresponding complex exponentials give the real solutions of the related constant coefficient homogeneous differential equation: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=