Unit 2: Complex Functions and Series
Table of Contents
Exponential Function (Complex Arguments)
We extend the definition of `eˣ` to complex numbers `z = x + iy`.
Definition
Using Euler's formula, we define the exponential function as:
eᶻ = eˣ⁺ⁱʸ = eˣ · eⁱʸ = eˣ(cos y + i sin y)
Properties
- Periodicity: The complex exponential function is periodic with a purely imaginary period of 2πi.
`eᶻ⁺²πⁱ = eᶻ · e²πⁱ = eᶻ(cos(2π) + i sin(2π)) = eᶻ(1 + 0) = eᶻ` - Modulus: `|eᶻ| = |eˣ(cos y + i sin y)| = |eˣ| · |cos y + i sin y| = eˣ · 1 = eˣ`
- Argument: `arg(eᶻ) = y` (or `y + 2kπ`)
Logarithmic Function (Complex Arguments)
The complex logarithm `w = log(z)` is the inverse of the exponential function. It is defined by `eʷ = z`.
Definition
Let `z = reⁱᶿ` and `w = u + iv`.
Then `eʷ = z` becomes `eᵘ⁺ⁱᵛ = reⁱᶿ`, which is `eᵘ(cos v + i sin v) = r(cos θ + i sin θ)`.
Comparing modulus and argument:
- `eᵘ = r` => `u = ln(r) = ln(|z|)`
- `v = θ`
However, since `θ` can be `θ + 2kπ` for any integer k, `v` can be `θ + 2kπ`.
The general logarithm is: log(z) = ln(r) + i(θ + 2kπ) = ln(|z|) + i · arg(z)
Principal Value
The principal value of the logarithm, denoted `Log(z)`, is the value when `k = 0` and the argument `θ` is in the principal range `(-π, π]`.
Log(z) = ln(r) + iθ, where `θ = Arg(z)`
Gregory's Series
Gregory's series is a power series expansion for the inverse tangent function.
Statement
If `θ` is an angle such that -π/4 ≤ θ ≤ π/4, then: θ = tan θ - (tan³θ / 3) + (tan⁵θ / 5) - (tan⁷θ / 7) + ...
By letting `x = tan θ`, we get the more common form:
tan⁻¹(x) = x - (x³ / 3) + (x⁵ / 5) - (x⁷ / 7) + ...
This is valid for -1 ≤ x ≤ 1.
Application: Calculating π
We can use this series to find the value of π. A common method is to set `x = 1`:
`tan⁻¹(1) = 1 - 1/3 + 1/5 - 1/7 + ...`
Since `tan⁻¹(1) = π/4`, we get:
π/4 = 1 - 1/3 + 1/5 - 1/7 + ... (Leibniz formula for π)Hyperbolic Functions
Hyperbolic functions are analogues of trigonometric functions, but defined using the hyperbola `x² - y² = 1` instead of the circle `x² + y² = 1`.
Definitions (in terms of exponentials)
Hyperbolic Sine: sinh(x) = (eˣ - e⁻ˣ) / 2
Hyperbolic Cosine: cosh(x) = (eˣ + e⁻ˣ) / 2
Hyperbolic Tangent: tanh(x) = sinh(x) / cosh(x) = (eˣ - e⁻ˣ) / (eˣ + e⁻ˣ)
Key Identity
Similar to `cos²x + sin²x = 1`, the key identity for hyperbolic functions is:
cosh²(x) - sinh²(x) = 1
Osborn's Rule (Relation to Trig Functions)
We can relate hyperbolic and trigonometric functions using complex arguments:
cos(ix) = (eⁱ(ⁱˣ) + e⁻ⁱ(ⁱˣ)) / 2 = (e⁻ˣ + eˣ) / 2 = cosh(x)sin(ix) = (eⁱ(ⁱˣ) - e⁻ⁱ(ⁱˣ)) / (2i) = (e⁻ˣ - eˣ) / (2i) = i(eˣ - e⁻ˣ) / (2i * i) = i(eˣ - e⁻ˣ) / (-2) = i sinh(x)
cos(ix) = cosh(x)
sin(ix) = i sinh(x)
cosh(ix) = cos(x)
sinh(ix) = i sin(x)
Summation of Trigonometric Series
A common method for summing trigonometric series is the C + iS method.
Method
- Let the series you want to sum be `C` (for a cosine series) or `S` (for a sine series).
- Create the corresponding `S` or `C` series.
- Combine them into a complex series: `C + iS`.
- Use Euler's formula `cos(nθ) + i sin(nθ) = eⁱⁿᶿ` to rewrite the complex series.
- The series will now be a standard geometric series (or related series). Sum this series.
- The result will be a complex number. Simplify it into the form `A + iB`.
- Equate the real and imaginary parts: `C = A` and `S = B`.
Example: Sum of a Cosine Series (Geometric)
Find the sum `C = 1 + r cos θ + r²cos(2θ) + ... + rⁿ⁻¹cos((n-1)θ)`.
- Let `S = r sin θ + r²sin(2θ) + ... + rⁿ⁻¹sin((n-1)θ)`.
- Consider `C + iS = 1 + r(cos θ + i sin θ) + r²(cos(2θ) + i sin(2θ)) + ...`
- `C + iS = 1 + r eⁱᶿ + r²eⁱ²ᶿ + ...`
- This is a geometric series with first term `a = 1` and common ratio `x = reⁱᶿ`.
- Sum = `a(1 - xⁿ) / (1 - x)` = `(1 - (reⁱᶿ)ⁿ) / (1 - reⁱᶿ)` = `(1 - rⁿeⁱⁿᶿ) / (1 - reⁱᶿ)`
- `C + iS = (1 - rⁿ(cos(nθ) + i sin(nθ))) / (1 - r(cos θ + i sin θ))`
- The final step (which is algebraically intensive) is to multiply the numerator and denominator by the conjugate of the denominator, and then separate the real part (`C`) from the imaginary part (`S`).