How can you determine the exact value of $$\int_{0}^{\pi/2} \sqrt{\sin{\theta}} \, d \theta$$ or $$\int_0^1 \ln^2{x} \ln^2{(1-x)} \, dx$$ by hand? What about $$ \int_0^{\infty} e^{-x^2} \ln{x} \, dx$$ or other similar integrals? The answer lies in series expansions and some really wacky special functions. We will go from infinite product expansions of entire functions to the Gamma function, and learn how this can be applied to things you didn't know you could integrate. Learn to evaluate stuff that doesn't even appear to converge -- and how to beat your computer to it! Along the way, you'll find out what sorts of implications the Riemann hypothesis has for the prime numbers, why $$1+2+3+4+5+\cdots = -1/12$$, what the volume of an n-dimensional hypersphere is, and how to get rid of infinity when you need him out of your calculations.

How can you determine the exact value of $$\int_{0}^{\pi/2} \sqrt{\sin{\theta}} \, d \theta$$ or $$\int_0^1 \ln^2{x} \ln^2{(1-x)} \, dx$$ by hand? What about $$ \int_0^{\infty} e^{-x^2} \ln{x} \, dx$$ or other similar integrals? The answer lies in series expansions and some really wacky special functions. We will go from infinite product expansions of entire functions to the Gamma function, and learn how this can be applied to things you didn't know you could integrate. Learn to evaluate stuff that doesn't even appear to converge -- and how to beat your computer to it! Along the way, you'll find out what sorts of implications the Riemann hypothesis has for the prime numbers, why $$1+2+3+4+5+\cdots = -1/12$$, what the volume of an n-dimensional hypersphere is, and how to get rid of infinity when you need him out of your calculations.

How can you determine the exact value of $$\int_{0}^{\pi/2} \sqrt{\sin{\theta}} \, d \theta$$ or $$\int_0^1 \ln^2{x} \ln^2{(1-x)} \, dx$$ by hand? What about $$ \int_0^{\infty} e^{-x^2} \ln{x} \, dx$$ or other similar integrals? The answer lies in series expansions and some really wacky special functions. We will go from infinite product expansions of entire functions to the Gamma function, and learn how this can be applied to things you didn't know you could integrate. Learn to evaluate stuff that doesn't even appear to converge -- and learn to beat your computer to it! Along the way, you'll find out what sorts of implications the Riemann hypothesis has for the prime numbers, why $$1+2+3+4+5+\cdots = -1/12$$, what the volume of an n-dimensional hypersphere is, and how to get rid of infinity when you need him out of your calculations.

How can you determine the exact value of $$\int_{0}^{\pi/2} \sqrt{\sin{\theta}} \, d \theta$$ or $$\int_0^1 \ln^2{x} \ln^2{(1-x)} \, dx$$ by hand? What about $$ \int_0^{\infty} \frac{x^2}{2^x - 1} \, dx$$ or other similar integrals? The answer lies in series expansions and some really wacky special functions. We will go from infinite product expansions of entire functions to the Gamma function, and learn how this can be applied to things you didn't know you could integrate. Learn to evaluate stuff that doesn't even appear to converge -- and learn to beat your computer to it! Along the way, you'll find out what sorts of implications the Riemann hypothesis has for the prime numbers, why the sum of all the positive integers is $$-1/12$$, what the volume of an n-dimensional hypersphere is, and how to get rid of infinity when you need him out of your calculations.