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In 1757, V. Riccati first recorded the generalizations of the hyperbolic functions defined by F_(n,r)^alpha(x)=sum_(k=0)^infty(alpha^k)/((nk+r)!)x^(nk+r), (1) for r=0, ..., ...

Let R[z]>0, 0<=alpha,beta<=1, and Lambda(alpha,beta,z)=sum_(r=0)^infty[lambda((r+alpha)z-ibeta)+lambda((r+1-alpha)z+ibeta)], (1) where lambda(x) = -ln(1-e^(-2pix)) (2) = ...

To solve the system of differential equations (dx)/(dt)=Ax(t)+p(t), (1) where A is a matrix and x and p are vectors, first consider the homogeneous case with p=0. The ...

Let all of the functions f_n(z)=sum_(k=0)^inftya_k^((n))(z-z_0)^k (1) with n=0, 1, 2, ..., be regular at least for |z-z_0|<r, and let F(z) = sum_(n=0)^(infty)f_n(z) (2) = (3) ...

A homework problem proposed in Steffi's math class in January 2003 asked students to prove that no ratio of two unequal numbers obtained by permuting all the digits 1, 2, ...

The metric g defined on a nonempty set X by g(x,x) = 0 (1) g(x,y) = 1 (2) if x!=y for all x,y in X. It follows that the open ball of radius r>0 and center at x_0 B(x_0,r)={x ...

Let A be a matrix and x and b vectors. Then the system Ax=b, x>=0 has no solution iff the system A^(T)y>=0, b^(T)y<0 has a solution, where y is a vector (Fang and Puthenpura ...

If g is a continuous function g(x) in [a,b] for all x in [a,b], then g has a fixed point in [a,b]. This can be proven by supposing that g(a)>=a g(b)<=b (1) g(a)-a>=0 ...

Let A = [B D; E C] (1) A^(-1) = [W X; Y Z], (2) where B and W are k×k matrices. Then det(Z)det(A)=det(B). (3) The proof follows from equating determinants on the two sides of ...

Let n be an elliptic pseudoprime associated with (E,P), and let n+1=2^sk with k odd and s>=0. Then n is a strong elliptic pseudoprime when either kP=0 (mod n) or 2^rkP=0 (mod ...