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If a function analytic at the origin has no singularities other than poles for finite x, and if we can choose a sequence of contours C_m about z=0 tending to infinity such ...

The unique nonnegative square root of a nonnegative real number. For example, the principal square root of 9 is 3, although both -3 and 3 are square roots of 9. The concept ...

The three circles theorem, also called Hadamard's three circles theorem (Edwards 2001, p. 187), states that if f is an analytic function in the annulus 0<r_1<|z|<r_2<infty, ...

There are at least two meanings of the term "total derivative" in mathematics. The first is as an alternate term for the convective derivative. The total derivative is the ...

e^(i(ntheta))=(e^(itheta))^n. (1) From the Euler formula it follows that cos(ntheta)+isin(ntheta)=(costheta+isintheta)^n. (2) A similar identity holds for the hyperbolic ...

The Jacobi theta functions are the elliptic analogs of the exponential function, and may be used to express the Jacobi elliptic functions. The theta functions are ...

Calabi-Yau spaces are important in string theory, where one model posits the geometry of the universe to consist of a ten-dimensional space of the form M×V, where M is a four ...

A cupolarotunda in which the bases are in the same orientation.

The Euler formula, sometimes also called the Euler identity (e.g., Trott 2004, p. 174), states e^(ix)=cosx+isinx, (1) where i is the imaginary unit. Note that Euler's ...

Given a first-order ordinary differential equation (dy)/(dx)=F(x,y), (1) if F(x,y) can be expressed using separation of variables as F(x,y)=X(x)Y(y), (2) then the equation ...