Complex integrals
About points...
We associate a certain number of points with each exercise.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
About difficulty...
We associate a certain difficulty with each exercise.
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
Question
Solution
Short
Video
\(\LaTeX\)
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Exercise:
Solve the following egral: _^pi fraccost-costddt
Solution:
Using costfrace^it+e^-it we get _^pi fraccost-costddt _^pi fracfrace^it+e^-it-e^it+e^-itddt Furthermore we can use the substitution z: e^it with ddtfracddziz in order to turn the egral o one over the unit disc using the parametrisation gammate^it. Thus we get: _^pi fracfrace^it+e^-it-e^it+e^-itddt frac_|z|fracz^+fracz^-leftz+fraczrightfracizddz We now need to reformulate the egrand in order to find the poles: fracz^+fracz^left-leftz+fraczrightrightz fracfracz^+z^fracz-z^-zfracz fracfracz^+z^z-z^- fracz^+z^z-z^- fracz^+-z^leftz^-fracz+right fracz^+-z^leftz-fracrightz- So we get three different poles: itemize item z order item zfrac order item z order itemize but only z frac are contained in the unit circle so then we continue by calculating the corresponding residues: textRes_fracf lim_zrightarrow frac fracz^+-z^z- fracleftfracright^+-leftfracright^leftfrac-right frac textRes_f fraclim_zrightarrow leftfracdddd zright^ fracz^+-leftz-fracrightz- -frac In mary this gives: _^pi fraccost-costddt pi i fraci leftfrac-fracright fracpi
Solve the following egral: _^pi fraccost-costddt
Solution:
Using costfrace^it+e^-it we get _^pi fraccost-costddt _^pi fracfrace^it+e^-it-e^it+e^-itddt Furthermore we can use the substitution z: e^it with ddtfracddziz in order to turn the egral o one over the unit disc using the parametrisation gammate^it. Thus we get: _^pi fracfrace^it+e^-it-e^it+e^-itddt frac_|z|fracz^+fracz^-leftz+fraczrightfracizddz We now need to reformulate the egrand in order to find the poles: fracz^+fracz^left-leftz+fraczrightrightz fracfracz^+z^fracz-z^-zfracz fracfracz^+z^z-z^- fracz^+z^z-z^- fracz^+-z^leftz^-fracz+right fracz^+-z^leftz-fracrightz- So we get three different poles: itemize item z order item zfrac order item z order itemize but only z frac are contained in the unit circle so then we continue by calculating the corresponding residues: textRes_fracf lim_zrightarrow frac fracz^+-z^z- fracleftfracright^+-leftfracright^leftfrac-right frac textRes_f fraclim_zrightarrow leftfracdddd zright^ fracz^+-leftz-fracrightz- -frac In mary this gives: _^pi fraccost-costddt pi i fraci leftfrac-fracright fracpi
Meta Information
Exercise:
Solve the following egral: _^pi fraccost-costddt
Solution:
Using costfrace^it+e^-it we get _^pi fraccost-costddt _^pi fracfrace^it+e^-it-e^it+e^-itddt Furthermore we can use the substitution z: e^it with ddtfracddziz in order to turn the egral o one over the unit disc using the parametrisation gammate^it. Thus we get: _^pi fracfrace^it+e^-it-e^it+e^-itddt frac_|z|fracz^+fracz^-leftz+fraczrightfracizddz We now need to reformulate the egrand in order to find the poles: fracz^+fracz^left-leftz+fraczrightrightz fracfracz^+z^fracz-z^-zfracz fracfracz^+z^z-z^- fracz^+z^z-z^- fracz^+-z^leftz^-fracz+right fracz^+-z^leftz-fracrightz- So we get three different poles: itemize item z order item zfrac order item z order itemize but only z frac are contained in the unit circle so then we continue by calculating the corresponding residues: textRes_fracf lim_zrightarrow frac fracz^+-z^z- fracleftfracright^+-leftfracright^leftfrac-right frac textRes_f fraclim_zrightarrow leftfracdddd zright^ fracz^+-leftz-fracrightz- -frac In mary this gives: _^pi fraccost-costddt pi i fraci leftfrac-fracright fracpi
Solve the following egral: _^pi fraccost-costddt
Solution:
Using costfrace^it+e^-it we get _^pi fraccost-costddt _^pi fracfrace^it+e^-it-e^it+e^-itddt Furthermore we can use the substitution z: e^it with ddtfracddziz in order to turn the egral o one over the unit disc using the parametrisation gammate^it. Thus we get: _^pi fracfrace^it+e^-it-e^it+e^-itddt frac_|z|fracz^+fracz^-leftz+fraczrightfracizddz We now need to reformulate the egrand in order to find the poles: fracz^+fracz^left-leftz+fraczrightrightz fracfracz^+z^fracz-z^-zfracz fracfracz^+z^z-z^- fracz^+z^z-z^- fracz^+-z^leftz^-fracz+right fracz^+-z^leftz-fracrightz- So we get three different poles: itemize item z order item zfrac order item z order itemize but only z frac are contained in the unit circle so then we continue by calculating the corresponding residues: textRes_fracf lim_zrightarrow frac fracz^+-z^z- fracleftfracright^+-leftfracright^leftfrac-right frac textRes_f fraclim_zrightarrow leftfracdddd zright^ fracz^+-leftz-fracrightz- -frac In mary this gives: _^pi fraccost-costddt pi i fraci leftfrac-fracright fracpi
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