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R(p,tau) = int_(-infty)^inftyint_(-infty)^infty[1/(sigmasqrt(2pi))e^(-(x^2+y^2)/(2sigma^2))]delta[y-(tau+px)]dydx (1) = ...

The amazing identity for all theta, where Gamma(z) is the gamma function. Equating coefficients of theta^0, theta^4, and theta^8 gives some amazing identities for the ...

The two-argument Ramanujan function is defined by phi(a,n) = 1+2sum_(k=1)^(n)1/((ak)^3-ak) (1) = 1-1/a(H_(-1/a)+H_(1/a)+2H_n-H_(n-1/a)-H_(n+1/a)). (2) The one-argument ...

A sum which includes both the Jacobi triple product and the q-binomial theorem as special cases. Ramanujan's sum is ...

int_0^inftycos(2zt)sech(pit)dt=1/2sechz for |I[z]|<pi/2. A related integral is int_0^inftycosh(2zt)sech(pit)dt=1/2secz for |R[z]|<pi/2.

sum_(n=0)^(infty)(-1)^n[((2n-1)!!)/((2n)!!)]^3 = 1-(1/2)^3+((1·3)/(2·4))^3+... (1) = _3F_2(1/2,1/2,1/2; 1,1;-1) (2) = [_2F_1(1/4,1/4; 1;-1)]^2 (3) = ...

int_(-infty)^infty(J_(mu+xi)(x))/(x^(mu+xi))(J_(nu-xi)(y))/(y^(nu-xi))e^(itxi)dxi =[(2cos(1/2t))/(x^2e^(-it/2)+y^2e^(it/2))]^((mu+nu)/2) ...

For a point y in Y, with f(y)=x, the ramification index of f at y is a positive integer e_y such that there is some open neighborhood U of y so that x has only one preimage ...

A type of cusp as illustrated above for the curve x^4+x^2y^2-2x^2y-xy^2+y^2=0.

The fractal-like figure obtained by performing the same iteration as for the Mandelbrot set, but adding a random component R, z_(n+1)=z_n^2+c+R. In the above plot, ...