Last modified on 22 November 2014, at 20:03

Exponential function

Plots of the Exponential Function over the Complex PlaneEdit

The exponential function is more complicated in the complex plane. On the real axis, the real part follow the expected exponential shape, and the imaginary part is identically zero. However, as the imaginary part changes, the exponential varies sinusoidally, with a period of 2π in the imaginary direction.

Real PartEdit

z=\operatorname{Re} \left (\exp \left( x + i y \right)\right)

In this plot, you can see the main branch on the real axis (y=0), and part of each branch on either side. This plot runs from -2π..2π in the y-direction and from -3..3 in the x-direction.

Exponential Function (Real Part).png
Exponential Function (Real Part) Density.png

The colour in the density plot (right) runs from red (large negative values), through green (nearly zero) to blue (large positive values) Please note that the this colour scheme is non-linear (it is based on the Arctan function and therefore emphaseses changes near zero more), and thus a given change in hue does NOT necessarily reflect a similar change in value, although at the same magnitude, the changes should be identical.

The plot below shows the real part of the exponential function as the operand approaches infinity. This plot is given by:

z=\operatorname{Re} \left (\exp \left( \frac{1}{x + i y} \right)\right)
Exponential Function (Real Part at Infinity).png Exponential Function (Real Part at Infinity) Density.png

The plot below shows the absolute value of the real part as the operand approaches infinity:

z=\bigg|\operatorname{Re} \left (\exp \left( \frac{1}{x + i y} \right)\right) \bigg|

The colouring of the density plots is different to the graphs above. It runs from green (small), through blue and red to yellow (large). It is highly non-linear and changes near zero give a larger hue shift than large values.

Exponential Function (Abs Real Part at Infinity).png Exponential Function (AbsReal Part at Infinity) Density.png


Imaginary PartEdit

z=\operatorname{Im} \left (\exp \left( x + i y \right)\right)
Exponential Function (Imag Part).png Exponential Function (Imag Part) Density.png

The plot below shows the imaginary part of the exponetial function as the operand approaches infinity. This plot is given by:

z=\operatorname{Im} \left (\exp \left( \frac{1}{x + i y} \right)\right)

Note that the orientation of the surface graph below is different to provide a better view of the structure of the function.

Exponential Function (Imag Part at Infinity).png Exponential Function (Imag Part at Infinity) Density.png

The plot below shows the absolute value of the imaginary part as the operand approaches infinity:

z=\bigg|\operatorname{Im} \left (\exp \left( \frac{1}{x + i y} \right)\right) \bigg|

The colouring of the density plots is different to the graphs above. It runs from green (small), through blue and red to yellow (large). It is highly non-linear and changes near zero give a larger hue shift than large values.

Exponential Function (Abs Imag Part at Infinity).png Exponential Function (Abs Imag Part at Infinity) Density.png

ModulusEdit

z=| \exp \left( x + i y \right)|

The modulus of the value is the same as the value of the exponential of just the real part of the operand. Since the real part behaves as a cosine across the complex plane, and the imaginary part like a sine, the modulus which is related to the real part squared plus the imaginary part squared. As \sin^2x + \cos^2x =1, it makes sense that the modulus is invariant in the imaginary direction.

Exponential Function Modulus.png 400px

ArgumentEdit

z=\arg \left (\exp \left( x + i y \right)\right)

The argument of the exponential function is simply the argument of the operand. The density plot has a linear colour function (an equal change in colour is representative of an equal change in the value of the point). Red is lowest, green is zero and blue is highest.

Exponential Function Argument.png 400px

Taylor Series ApproximationsEdit

Real Part, Main BranchEdit

\mathrm{e}^{x} = \sum^{\infin}_{n=0} \frac{x^n}{n!}