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Complex Functions - Basic Mappings
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f(z) = z + c, (where c is a complex constant)
Translates the complex plane by the constant c.
f(z) = z * c, (where c is a non-zero complex constant)
Rotates and dilates the complex plane by the constant c.
f(z) = \sin(z)
Applies the sine function to both the real and imaginary parts, causing horizontal and vertical stretching and compressing.
f(z) = e^z
Maps the complex plane onto the punctured complex plane excluding the origin, with exponential stretching and infinite wrapping.
f(z) = |z|^2
Maps z to the square of its magnitude, essentially stretching the plane away from the origin and squaring the distance from the origin for every point.
f(z) = \sqrt{z}
Takes the square root of the magnitude and halves the angle of z.
f(z) = \log(z)
Maps the complex plane to horizontal strips corresponding to the magnitude of z and the angle to vertical lines.
f(z) = z - z^*
Maps each complex number z to twice its imaginary part, effectively projecting the complex plane onto the imaginary axis.
f(z) = \frac{1}{z}, z \neq 0
Inverts every nonzero point in the complex plane about the unit circle and reflects it across the real axis.
f(z) = z + z^*
Maps each complex number z to twice its real part, effectively projecting the complex plane onto the real axis.
f(z) = \Im(z), (\Im(z) denotes the imaginary part of z)
Projects the complex plane onto the imaginary axis.
f(z) = z^2
Squares the magnitude and doubles the angle of z.
f(z) = e^{iz}
Maps the complex plane onto the unit circle.
f(z) = |z|
Takes the absolute value of z, projecting the complex plane onto the positive real axis.
f(z) = z \cdot e^{i\theta}, (where \theta is a real constant)
Rotates the complex plane by the angle \theta.
f(z) = \Re(z), (\Re(z) denotes the real part of z)
Projects the complex plane onto the real axis.
f(z) = \cos(z)
Applies the cosine function to both the real and imaginary parts, resulting in a pattern similar to the sine function but shifted by radians.
f(z) = \tan(z)
Combines complex sine and cosine functions, resulting in repetitive rectangular patterns with singularities.
f(z) = \cosh(z)
Applies hyperbolic cosine to the complex plane, leading to unbounded exponential growth away from the imaginary axis.
f(z) = \overline{z}
Reflects the complex plane across the real axis.
f(z) = \frac{1}{z+c}, (where c is a complex constant and z \neq -c)
Performs an inversion about a circle centered at -c, reflecting the plane across the real axis.
f(z) = -z
Performs a 180-degree rotation around the origin.
f(z) = \tanh(z)
Applies hyperbolic tangent to the complex plane, squeezing it onto the strip between -1 and 1 along the real axis.
f(z) = \frac{1}{z^2}, z \neq 0
Inverts the square of the magnitude of z and subtracts twice the angle of z
f(z) = z^{-1}
Inverts the magnitude and negates the angle of z, equivalent to the mapping by .
f(z) = \text{arg}(z), (\text{arg}(z) gives the angle of z)
Projects each non-zero complex number to its argument on the unit circle.
f(z) = \text{sgn}(z), (where \text{sgn} denotes the signum function)
Maps every non-zero complex number to a point on the unit circle corresponding to its direction from the origin.
f(z) = iz
Rotates the complex plane 90 degrees counterclockwise.
f(z) = z^n, n>0
Raises the magnitude to the nth power and multiplies the angle by n.
f(z) = z^3
Cubes the magnitude and triples the angle of z.
f(z) = \sinh(z)
Applies hyperbolic sine to the complex plane causing a double-sided exponential growth pattern along the real axis.
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