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1.3 The Exponential Function

1.3.1 A Power Series for {e}^{x}

We already know the exponential function {e}^{x} for real x. We want to generalise it to complex arguments z because it is the most important function in your (physics) life! Let’s have a look at {e}^{x} again. We can define {e}^{x} as follows:

The exponential function f(x) =\mathop{ exp}\nolimits (x) is the unique solution of the first order differential equation f'(x) = f(x),\quad f(x = 0) = 1.

This describes, for example, the increase of the number f of animals with time x (strange choice of variable, I know) at unit rate 1, if at time x = 0 there was one animal.

Task: 1. Think of other, better examples.
2. Can you define for {e}^{−x}?
In this case, there are more and better examples, se last semester’s course notes.

Now, the solution of the differential equation above is f(x) = {e}^{x}, but can we express this in a different manner? Suppose you had no \mathop{exp}\nolimits –button on your scientific calculator, and you were a survivor on a remote planet with the task to reconstruct mathematics and physics as a part of personkind’s knowledge, how would you calculate {e}^{x} ? Let’s try to write f(x) as a ‘polynomial’

\begin{eqnarray} f(x) = 1 + {a}_{1}x + {a}_{2}{x}^{2} + {a}_{ 3}{x}^{3} + ...,& & %&(1.30) \\ \end{eqnarray}

where we have to determine the constants {a}_{1}, {a}_{2},... from the differential equation:

\begin{eqnarray} f'(x)& =& {a}_{1} + 2{a}_{2}x + 3{a}_{3}{x}^{2} + 4{a}_{ 4}{x}^{3} + ...%& \\ f(x)& =& 1 + {a}_{1}x + {a}_{2}{x}^{2} + {a}_{ 3}{x}^{3} %&(1.31) \\ \end{eqnarray}

We now equate terms of the same power in x, since different powers vary in different ways as x changes, so we can disentagle the uinfinite series (cf. equality of polynomials)

\begin{eqnarray} ⇝ {a}_{1}& =& 1,\quad {a}_{2} = {1\over 1 ⋅ 2},\quad {a}_{3} = {1\over 1 ⋅ 2 ⋅ 3},..%&(1.32) \\ \end{eqnarray}

We recognise that the general form of this power series is

\begin{eqnarray} f(x) =\mathop{ exp}\nolimits (x) = 1 + {1\over 1!}x + {1\over 2!}{x}^{2} + {1\over 3!}{x}^{3} + ...,\quad n! := 1 ⋅ 2 ⋅ 3... ⋅ n.& & %&(1.33) \\ \end{eqnarray}

The symbol n! := 1 ⋅ 2 ⋅ 3... ⋅ n is called the factorial. One defines 0! = 1. We write the equation for f(x) in another, more condensed and elegant form

\begin{eqnarray} \mathop{exp}\nolimits (x) ={ \mathop{∑ }}_{n=0}^{∞}{{x}^{n}\over n!} & & %&(1.34) \\ \end{eqnarray}

You have to remember this formula throughout your whole life. Now, we generalise this ‘all–your–life’ formula to complex numbers z,

\begin{eqnarray} \mathop{exp}\nolimits (z) ={ \mathop{∑ }}_{n=0}^{∞}{{z}^{n}\over n!} \quad .& & %&(1.35) \\ \end{eqnarray}

1.3.2 Power Series for \mathop{sin}\nolimits (x) and \mathop{cos}\nolimits (x)

Next to the exponential, certainly \mathop{sin}\nolimits and \mathop{cos}\nolimits belong to some of the most important functions in physics. In much the same spirit as above, we can define them by their differential equations, and derive an expression in terms of a series. Here we state the result, that can be used to define \mathop{sin}\nolimits and \mathop{cos}\nolimits :

\begin{eqnarray} \mathop{sin}\nolimits (x)& :=& {\mathop{∑ }}_{n=0}^{∞}{{(−1)}^{n}{x}^{2n+1}\over (2n + 1)!} = x −{{x}^{3}\over 3!} + {{x}^{5}\over 5!} −{{x}^{7}\over 7!} + ...%& \\ \mathop{cos}\nolimits (x)& :=& {\mathop{∑ }}_{n=0}^{∞}{{(−1)}^{n}{x}^{2n}\over (2n)!} = 1 −{{x}^{2}\over 2!} + {{x}^{4}\over 4!} −{{x}^{6}\over 6!} + ... %&(1.36) \\ \end{eqnarray}

task: Calculate the derivatives of \mathop{sin}\nolimits (x) and \mathop{cos}\nolimits (x) from the series definition above and check that it is consistent! (i.e., \mathop{sin}\nolimits '(x) =\mathop{ cos}\nolimits (x) and \mathop{cos}\nolimits '(x) = −\mathop{sin}\nolimits (x).

Now we are in a position to test our series for the complex exponential,

\begin{eqnarray} \mathop{exp}\nolimits (z) ={ \mathop{∑ }}_{n=0}^{∞}{{z}^{n}\over n!} & & %&(1.37) \\ \end{eqnarray}

We use it for a purely imaginary variable,

\begin{eqnarray} z = x + iy = iy,\quad (x = 0),& & %&(1.38) \\ \end{eqnarray}

where y is real (remember that now that we have both real and complex numbers, one always has to state which type of number one is talking about). Remember

\begin{eqnarray}{ i}^{0} = 1,\quad {i}^{1} = i,\quad {i}^{2} = −1,\quad {i}^{3} = −i,\quad {i}^{4} = 1,\quad {i}^{5} = i,...& & %&(1.39) \\ \end{eqnarray}

This lead to

\begin{eqnarray} \mathop{exp}\nolimits (iy) = 1 + {iy\over 1!} −{{y}^{2}\over 2!} −{i{y}^{3}\over 3!} + {{y}^{4}\over 4!} + {i{y}^{5}\over 5!} − ...& & %&(1.40) \\ \end{eqnarray}

The terms are alternately real and imaginary, so we cansplit the series

\begin{eqnarray} \mathop{exp}\nolimits (iy) = \left (1 −{{y}^{2}\over 2!} + {{y}^{4}\over 4!} − ...\right ) + i\left ( {y\over 1!} −{{y}^{3}\over 3!} + {{y}^{5}\over 5!} + ...\right ).& & %&(1.41) \\ \end{eqnarray}

We compare the right and side of this equation with the series for \mathop{sin}\nolimits (y) and \mathop{cos}\nolimits (y), we find the important Euler’s Formula

\begin{eqnarray} \mathop{exp}\nolimits (iy) =\mathop{ cos}\nolimits (y) + i\mathop{sin}\nolimits (y),\quad y\text{ real}.& & %&(1.42) \\ \end{eqnarray}

Using Euler’s formula for the variable y = θ (angle in our polar representation), we find for any complex number z the representation

\begin{eqnarray} z = x + iy = r[\mathop{cos}\nolimits (θ) + i\mathop{sin}\nolimits (θ)] = r\mathop{exp}\nolimits (iθ).& & %&(1.43) \\ \end{eqnarray}

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