60 seconds to solve as many complex integrals as possible. All residues should be in the region bound by the simple closed curve you integrate over, and the residues are usually clean rational numbers. Your score increases 1 for each correct integral, and with accuracy of over 80%, you “pass”, meaning if you score higher than the WR with 80% or above accuracy, you can get the new world record. Turbomode (and turbowarp) heavily recommended. My best is accuracy 100%, score 19, which is about 3.53 seconds per integral. Can you beat it? Ranks: Newbie: 0~1-> You're just starting out. Maybe you only got 1 point because you were lucky, who knows? Rookie: 2~3-> You're probably solving integrals at the 'standard' pace for an undergraduate student who doesn't worry much about mental math. Medium: 4~7-> So you're kind of interested in being the fastest guy in the room, huh? Fast: 8~10-> You're good at residue calculus of rational functions. Like, really good. You actually need to have formulas, tricks, etc. specifically meant for mental complex integration at this point. Lightning: 11~15-> Ok that is insane. You can be very proud of yourself, because you're solving an integral in 4~6 seconds on average, which is monstrous. Godspeed: 16+-> Um... What? I don't know how many people in the world right now could get godspeed, there's a lot of luck and noise that could prevent you from consistently getting it too. I plan to update it to make more variation in difficulty and stuff. HOW TO CALCULATE COMPLEX INTEGRALS For rational functions quadratic divided by quadratic, in the form (az^2+bz+c)/((z-d)(z-e)), Given all of the residues are in the region bounded by the simple closed curve, calculating the residues and simplifying yields the value a(d+e)+b. In fact, that d+e term is awesome- Vieta’s formula lets you compute that term without even factorizing the denominator, meaning messy roots don’t matter! So multiply by 2iπ to get the answer. Example problem. Find the value of the contour integral ∲_C (2z^2-4z+3)/((z-2)(z+4))dz along the curve C: |z|=11. Solution: Our two poles are at 2 and -4, both within the radius 11 circle on the complex plane. So we use the formula: 2(2-4)-4=-8 And our answer is -8*2iπ=-16iπ. I don’t actually think these kinds of “tricks” are learned at the undergraduate level since it shifts from basic computation to depth and proof understanding, but these kinds of trick formulas still exist, and it’s fun to discover/derive them yourself. This one is a fun one which only requires the residue theorem (or Cauchy’s integral theorem if you really want to do some mental gymnastics) to derive.These quadratic integrals actually have the same value as the integral of (az+b)/(z-(d+e)), along a simple closed curve that has its simple pole inside it, which is linear divided by linear. In fact, all rational functions in the form polynomial/quadratic can be expressed in a very similar way to a(d+e)+b, and surprisingly, all of these residues sum to an integer! I’ll leave the proof for the quadratic/quadratic and generalization/integer-ness of this result as an exercise for people who are keen on figuring out why this formula works (I don’t think it’s on any website or book as like I said, it’s more a special trick than a general formula, so good luck figuring it out lol), and maybe you can go further than I did by generalizing to cubic/quartic denominators and more. As for people who aren’t quite at that level yet… At least now you know a special case of the residue theorem and can apply it to mental math lol. You can solve quite a few GRE math subject test complex analysis problems mentally now. I'm starting a series I guess. So far, I have: -Transcendental Functions Trainer (train mental transcendental functions) -Complex Analysis Trainer (train mental residue calculus) Upcoming: -Linear Algebra Trainer (train mental eigenvalue computation) -Numerical Analysis Trainer (train finding roots of equations numerically) -Number Theory Trainer (train prime factorizing numbers)
