Complex Numbers

new to Analytica 4.5

Typing complex literals

A complex number has a real part and an imaginary part and is written as, e.g., 3.4 + 2.3j, where 3.4 is the real part and 2.3 is the imaginary part. If there is no imaginary part, then it is a real number.

To type the imaginary part, append a lower case i or lower case j to the number. For example, 1i or 1j is the square root of -1. You must have a digit before the i or j, so you can't type just i -- that would refer to a variable named i. You also cannot put a space between the numbers and the i or j. Numeric suffixes come before the i or j, so you would write 3.4Kj, 1.2mj or 1.2e-3j.

If you have a variable, x, that holds a complex number, and you want to use its value for the imaginary part, it does not work to type xj -- that would refer to a variable named xj. You need to type x * 1j.

Complex numbers display in result tables as the real part plus or minus the complex part, and using a small j after the complex part, e.g., -5 - 3j, 7.2 + 0.1j. The real and complex parts each use the prevailing number format.

When complex numbers appear on graphs, the real part is graphed. See #Graphing below.

Precision and range

The real and imaginary parts of a complex number each have approximately 7 decimal digits of precision and can range from -3.4e+38 to 3.4e+38 and as small as 3.4e-38. This is less precision than is used for a real number in Analytica, which has roughly 15 decimal digits of precision and a range from -1.7e+308 to 1.7e+308, and down to 1.7e-308. One complex number takes up the same amount of memory as one real number (the complex number stores two 32-bit floats, whereas the a real number stores one 64-bit float).

Accessing the parts of a complex number

To extract the real or imaginary part of a complex number, as a real number, use RealPart(x) or ImPart(x).

A complex number can also be written in polar form as [math]\displaystyle{ A e^{\theta j} }[/math], where A is the magnitude and [math]\displaystyle{ \theta }[/math] is the phase. To retrieve the magnitude use Abs(x). The phase can be extracted either in degrees or radians using DegreesPart(x) or RadiansPart(x).

To construct a complex number from its magnitude, A, and phase in radians, r, use A * Exp(r * 1j). Given a phase in degrees, d, use A * Exp(Radians(d) * 1j).

Arithmetic operations

The operations of +, -, * and / are standard from any textbook:

• [math]\displaystyle{ (a + b j) + (c + d j) = (a+c) + (b+d) j }[/math]
• [math]\displaystyle{ (a + b j) - (c + d j) = (a-c) + (b-d) j }[/math]
• [math]\displaystyle{ (a + b j) * (c + d j) = (ac - bd) + (bc + ad) j }[/math]
• [math]\displaystyle{ (a + b j) / (c + d j) = {{ac + bd}\over{c^2+d^2}} + {{bc-ad}\over{c^2+d^2}} j }[/math]

The conjugate of a complex number is RealPart(x) - ImPart(x).

Exponentiation is trickier. It is, of course, the case that 1j^2 is -1. But the case where a complex number is raised to a complex power is not at all straightforward. Need to write this part..

See Also

Accessing the parts of a complex number:

• RealPart, ImPart, Abs, DegreesPart, RadiansPart

Non-matrix functions that compute or return complex numbers:

• Sqrt, Ln, LogTen, Exp
• Sin, Cos, Tan, ArcSin, ArcCos, ArcTan

Matrix functions that handle complex matrices:

• Invert, SingularValueDecomp
• FFT, FFTInv