cis (mathematics)

Cis is a form of mathematical notation in which $\operatorname{cis}(x) = \cos(x) + i\sin(x),\,$, where cos is the cosine function, i is the imaginary unit and sin is the sine.

It was first coined by Irving Stringham in works such as Uniplanar Algebra (1893) and subsequently used by James Harkness and Frank Morley in their Introduction to the Theory of Analytic Functions (1898). It connects trigonometric functions with exponential functions in the complex plane via Euler's formula.

Relation to the complex exponential function

$e^{ix} = \cos(x) + i\sin(x)\,$,

$e^{-ix} = \cos(-x) + i\sin(-x) = \cos(x) - i\sin(x)\,$

$e^{i\pi} = -1\,$

$\cos(x) = \frac{e^{ix} + e^{-ix}}{2} \;$

$\sin(x) = \frac{e^{ix} - e^{-ix}}{2i} \;$

where i² = −1.

This can also be expressed using the following notation

$\operatorname{cis}(x) = \cos(x) + i\sin(x),\,$

i.e. "cis" abbreviates "cos + i sin".

Though at first glance this notation is redundant, being equivalent to e^ix, its use is rooted in several advantages.

Mathematical identities

Derivative

$\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{cis}(z) = ie^{iz}$

Integral

$\int\operatorname{cis}(z)\,\mathrm{d}z = -ie^{iz}$

Other properties

These follow directly from Euler's formula.

${\rm c\dot{\imath} s}(x+y)={\rm c\dot{\imath} s}(x)\,{\rm c\dot{\imath} s}(y)$

${\rm c\dot{\imath} s}(x-y)={{\rm c\dot{\imath} s}(x)\over{\rm c\dot{\imath} s}(y)}$

History

This notation was more common in the post-World-War-II era when typewriters were used to convey mathematical expressions.

Superscripts are both offset vertically and smaller than 'cis' or 'exp'; hence, they can be problematic even for hand-writing, for example, e^ix² versus cis(x²) versus exp(ix²). For many readers, cis(x²) is the clearest, easiest to read of the three.

The cis-notation is sometimes used to emphasize one method of viewing and dealing with a problem over another. The mathematics of trigonometry and exponentials are related but not exactly the same; exponential notation emphasizes the whole, whereas cis and cos + i sin notations emphasize the parts. This can be rhetorically useful to mathematicians and engineers when discussing this function, and further serve as a mnemonic (for cos + i sin).

The cis-notation is convenient for math students whose knowledge of trigonometry and complex numbers permit this notation, but whose conceptual understanding doesn't yet permit the notation e ^ix. As students learn concepts that build on prior knowledge, it is important not to force them into levels of math they are not yet prepared for: the proof that cis(x) = e^ix requires calculus, which the student may not have studied before they encountered the expression cos(x) + i sin(x).

Pedagogical use

In some contexts, the cis-notation may serve the pedagogical purpose of emphasizing that one has not yet proved that this is an exponential function. In doing trigonometry without complex numbers, one may prove the two identities

$\cos(x+y) = \cos(x)\cos(y) - \sin(x)\sin(y) = c_1 c_2 - s_1 s_2,\,$

$\sin(x+y) = \sin(x)\cos(y) + \cos(x)\sin(y) = s_1 c_2 + c_1 s_2.\,$

Similarly in treating multiplication of complex numbers (with no involvement of trigonometry), one may observe that the real and imaginary parts of the product of c₁ + is₁ and c₂ + is₂ are respectively

$c_1 c_2 - s_1 s_2,\,$

$s_1 c_2 + c_1 s_2.\,$

Thus one sees this same pattern arising in two disparate contexts:

• trigonometry without complex numbers, and
• complex numbers without trigonometry.

This coincidence can serve as a motivation for conjoining the two contexts and thereby discovering the trigonometric identity

$\operatorname{cis}(x+y) = \operatorname{cis}(x)\operatorname{cis}(y),\,$

and observing that this identity for cis of a sum is simpler than the identities for sin and cos of a sum. Having proved this identity, one can challenge the students to recall which familiar sort of function satisfies this same functional equation

$f(x+y) = f(x)f(y).\,$

The answer is exponential functions. That suggests that cis may be an exponential function

$\operatorname{cis}(x) = b^x.\,$

Then the question is: what is the base b? The definition of cis and the local behavior of sin and cos near zero suggest that

$\operatorname{cis}(0+dx) = \operatorname{cis}(0) + i\,dx,$

(where dx is an infinitesimal increment of x). Thus the rate of change at 0 is i, so the base should be eⁱ. Thus if this is an exponential function, then it must be

$\operatorname{cis}(x) = e^{ix}.$

See also

• De Moivre's formula
• Euler's formula
• Complex number
• Ptolemaios' theorem
• Phasor