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Let G be a subgroup of the modular group Gamma. Then an open subset R_G of the upper half-plane H is called a fundamental region of G if 1. No two distinct points of R_G are ...

A complex number z is said to be purely imaginary if it has no real part, i.e., R[z]=0. The term is often used in preference to the simpler "imaginary" in situations where z ...

The imaginary part I[z] of a complex number z=x+iy is the real number multiplying i, so I[x+iy]=y. In terms of z itself, I[z]=(z-z^_)/(2i), where z^_ is the complex conjugate ...

A pair of values x and y one or both of which is complex.

A complex function is said to be analytic on a region R if it is complex differentiable at every point in R. The terms holomorphic function, differentiable function, and ...

The real part R[z] of a complex number z=x+iy is the real number not multiplying i, so R[x+iy]=x. In terms of z itself, R[z]=1/2(z+z^_), where z^_ is the complex conjugate of ...

A Riemann surface is a surface-like configuration that covers the complex plane with several, and in general infinitely many, "sheets." These sheets can have very complicated ...

If a contour in the complex plane is curved such that it separates the increasing and decreasing sequences of poles, then ...

The E_n(x) function is defined by the integral E_n(x)=int_1^infty(e^(-xt)dt)/(t^n) (1) and is given by the Wolfram Language function ExpIntegralE[n, x]. Defining t=eta^(-1) ...

The identity function id(x) is the function id(x)=x which assigns every real number x to the same real number x. It is identical to the identity map. The identity function is ...