## Sunday, May 15, 2011

### Complex Number Operations (Part II)

For all the calculations, let z = a + bi

Exponential of a Complex Number

e^z = e^a (cos b + i sin b)

This calculation is fairly easy because it is relatively straight forward.

A sample RPN (Reverse Polish Notation) routine would look like:

With b on the y stack and a on the x stack:

e^x
STO 0 (or whatever register or variable you want as temporary storage)
x< >y (swap)
ENTER
SIN
RCL 0
×
R↓
COS
×
x< >y (swap)
R↓

Result: real(e^z) is on the x stack, while imag(e^z) is on the y stack

Natural Logarithm of a Complex Number

ln z = ln r + i θ

where r = √(x^2 + y^2) and θ = arctan(y/x)

Casio fx-115ES:
ln(abs(a + bi)) + i arg(a + bi)

Sharp EL-W516:
ln(abs(a + bi)) + i tan^-1(b÷a)

TI-36X Pro:
ln(abs(a + bi)) + i angle(a + bi)

With b on the y stack and a on the x stack, a sample RPN routine mat look like this:
→P (Rectangular to Polar conversion)
LN

Argument vs Arctangent (tan^-1)

Bear in mind that the argument (often labeled arg or angle for TI calulators) and the arctangent functions returns different angles. However the angles are accurate, recall the identity tan(x ± π) = tan(x).

Ranges of Calculator Functions:

angle/arg returns the angle between -π/2 and π/2 (-90° and 90°)

tan^-1 returns the angle between 0 and π (0° and 180°)

Sine of a Complex Number

There are two general formulas to calculate the sine of a complex number

(1) sin z = sin a cosh b + i cos a sinh b

(2) sin z = (e^iz - e^-iz)/(2i)

Formula (2) is good if your calculator can handle e^z. For the non-graphing calculators, I recommend using formula (1).

A sample RPN routine looks like this:

With b on the y stack and a on the x stack:

STO 0 (or A)
COS
x< >y
STO 1 (or B)
SINH
×
RCL 1 (or B)
COSH
RCL 0 (or A)
SIN
×

Cosine of a Complex Number

Calculating the cosine of a complex number is similar:

(1) cos z = cos a cosh b - i sin a sinh b

(2) cos z = (e^iz + e^-iz)/2

A sample RPN routine looks like this:

With b on the y stack and a on the x stack:

STO 0 (or A)
SIN
CHS (+/-)
x< >y
STO 1 (or B)
SINH
×
RCL 1 (or B)
COSH
RCL 0 (or A)
COS
×

The formulas are presented for completion.

ArcSine and ArcCosine

These formulas are presented for completion.

sin^-1 z = -i ln( i z ± √(1 - z^2))

cos^-1 z = -i ln(z ± i√(1 - z^2))

The principle angle is given by using +.

(staff.jccc.net)

Complex Power of a Complex Number

Let z and w be complex numbers, then:

z^w = e^(w ln z)

(HP 41C Math Pac)