MA30056: Complex Analysis

II.4. Examples of Holomorphic Functions

We want to get some geometric intuition of complexfunctions. Two methods we have already mentionedare to investigate the images/pre-images of simplecurves (like vertical and horizontal lines) under thefunction in question, or the level curves of the func-tion. If we now would like to see their graph infour-dimensional(!) space, we can refine the levelcurve method as follows: We use a colour palette tovisualise two of the four dimensions. We colourthe origin 0 white and colours get darker the greaterthe modulus of the complex number is (the point atinfinity is black). A colour corresponds1 to theargument of the complex number, e.g., the positivereal numbers are red-ish. In the following, for a givenfunction f , a point z C is coloured according toits value f(z) by the above palette, e.g., if we havef(z) = 1 for a point z then that point z is colouredred.

A different method is to study vector fields: Instead of colouring the z according to f(z), we attach the vectorf(z) = (Re f(z), Im f(z)) to the point z. Below is the result if we do this for f(z) = cos(z) (on the left) andf(z) = cos(z) (on the right).

1 Compare this with the earlier discussed level sets/curves: the level curve of constant modulus corresponds to the line ofthe same brightness/darkness, while the level curve of constant argument corresponds to the line of constant colour (whichmight, however, become brighter or darker).

The function z z2:

The image of the vertical (solid) and horizontal (dotted) linesunder f(z) = z2; they intersect perpendicularly.

Vector field (Re z2, Im z2) and colouring of the complex planefor z 7 z2.

Colouring and level curves of constant real (dotted) and imag-inary (solid) part for z 7 z2; they intersect perpendicularly.

Colouring and level curves of constant modulus (dotted) andargument (solid) for z 7 z2; they intersect perpendicularly.

The function z 7 z3:

The image of the vertical (solid) and horizontal (dotted) linesunder f(z) = z3; they intersect perpendicularly.

Vector field (Re z3, Im z3) and colouring of the complex planefor z 7 z3.

Colouring and level curves of constant real (dotted) and imag-inary (solid) part for z 7 z3; they intersect perpendicularly.

Colouring and level curves of constant modulus (dotted) andargument (solid) for z 7 z3; they intersect perpendicularly.

The complex exponential:

The image of the vertical (solid) and horizontal (dotted) linesunder f(z) = exp z; they intersect perpendicularly.

Vector field (Re exp z, Im exp z) and colouring of the complexplane for exp z.

Colouring and level curves of constant real (dotted) and imag-inary (solid) part for exp z; they intersect perpendicularly.

Colouring and level curves of constant modulus (dotted) andargument (solid) for exp z; they intersect perpendicularly.

The complex cosine:

The image of the vertical (solid) and horizontal (dotted) linesunder f(z) = cos z; they intersect perpendicularly.

Vector field (Re cos z, Im cos z) and colouring of the complexplane for cos z.

Colouring and level curves of constant real (dotted) and imag-inary (solid) part for cos z; they intersect perpendicularly.

Colouring and level curves of constant modulus (dotted) andargument (solid) for cos z; they intersect perpendicularly.

The complex sine:

The image of the vertical (solid) and horizontal (dotted) linesunder f(z) = sin z; they intersect perpendicularly.

Vector field (Re sin z, Im sin z) and colouring of the complexplane for sin z.

Colouring and level curves of constant real (dotted) and imag-inary (solid) part for sin z; they intersect perpendicularly.

Colouring and level curves of constant modulus (dotted) andargument (solid) for sin z; they intersect perpendicularly.

The complex hyperbolic cosine:

The image of the vertical (solid) and horizontal (dotted) linesunder f(z) = cosh z; they intersect perpendicularly.

Vector field (Re cosh z, Im cosh z) and colouring of the com-plex plane for cosh z.

Colouring and level curves of constant real (dotted) and imag-inary (solid) part for cosh z; they intersect perpendicularly.

Colouring and level curves of constant modulus (dotted) andargument (solid) for cosh z; they intersect perpendicularly.

The complex hyperbolic sine:

The image of the vertical (solid) and horizontal (dotted) linesunder f(z) = sinh z; they intersect perpendicularly.

Vector field (Re sinh z, Im sinh z) and colouring of the complexplane for sinh z.

Colouring and level curves of constant real (dotted) and imag-inary (solid) part for sinh z; they intersect perpendicularly.

Colouring and level curves of constant modulus (dotted) andargument (solid) for sinh z; they intersect perpendicularly.

Riemanns zeta function2 :

The image of the vertical (solid) and horizontal (dotted) linesunder f(z) = (z); they intersect perpendicularly.

Vector field (Re (z), Im (z)) and colouring of the complexplane for (z).

Colouring and level curves of constant real (dotted) and imag-inary (solid) part for (z); they intersect perpendicularly.

Colouring and level curves of constant modulus (dotted) andargument (solid) for (z); they intersect perpendicularly.

2 An article that uses level curves to study Riemanns zeta function in the critical strip can be found at

http://arxiv.org/abs/math.NT/0309433

http://arxiv.org/abs/math.NT/0309433