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A complex translation is a map of the form z|->z+a, where a is a fixed complex number, which corresponds to translation (shifting) of points the complex plane by the complex ...

The two-dimensional space consisting of the set of triples {(a,b,c):a,b,c in K, not all zero}, where triples which are scalar multiples of each other are identified.

The complex plane C with the origin removed, i.e., C-{0}. The punctured plane is sometimes denoted C^* (although this notation conflicts with that for the Riemann sphere C-*, ...

The portion of the complex plane z=x+iy with real part R[z]<0.

The portion of the complex plane z=x+iy with real part R[z]>0.

Complex infinity is an infinite number in the complex plane whose complex argument is unknown or undefined. Complex infinity may be returned by the Wolfram Language, where it ...

A complex rotation is a map of the form z|->ze^(itheta), where theta is a real number, which corresponds to counterclockwise rotation by theta radians about the origin of ...

A complex magnification is a map of the form z|->az, where a is a positive real number, which corresponds to magnification about the origin of points in the complex plane by ...

The portion of the complex plane {x+iy:x,y in (-infty,infty)} satisfying y=I[z]<0, i.e., {x+iy:x in (-infty,infty),y in (-infty,0)}

Let z=x+iy and f(z)=u(x,y)+iv(x,y) on some region G containing the point z_0. If f(z) satisfies the Cauchy-Riemann equations and has continuous first partial derivatives in ...