Elementary Complex Functions Exp Log Trig: Notes and FAQs

The complex exponential e^z = e^x(cos y + i sin y) extends the real exponential via Euler's formula e^{iy} = cos y + i sin y. Unlike the real version it is periodic with period 2πi and never zero. The complex logarithm inverts it: Log z = ln|z| + i Arg z (z ≠ 0). Complex trig functions are exponentials in disguise: sin z = (e^{iz} − e^{−iz})/(2i), cos z = (e^{iz} + e^{−iz})/2.

Oscillations and growth are inseparable in science: AC current, waves, and quantum phases all involve e^{i(something)}. These extensions unify trigonometry with exponentials and power phasor analysis, Fourier methods, and conformal mapping.

Key Concepts

• Complex exponential: e^{x+iy} = e^x(cos y + i sin y); modulus e^x, argument y.
• Periodicity: e^{z+2πi} = e^z; the plane wraps infinitely often around the punctured plane.
• Principal logarithm: Log z = ln|z| + i Arg z, Arg z in (−π, π]; the general log adds 2πik.
• Complex sine/cosine: defined via exponentials; sin²z + cos²z = 1 still holds.
• Unboundedness: |sin(iy)| = sinh|y| grows without bound; complex trig is not confined to [−1, 1].

Formulas

• e^{z_1+z_2} = e^{z_1}e^{z_2}; |e^z| = e^{Re z}; e^z ≠ 0 always.
• Log(zw) = Log z + Log w up to an integer multiple of 2πi (branch-cut price).
• sin(x+iy) = sin x cosh y + i cos x sinh y; cos(x+iy) = cos x cosh y − i sin x sinh y.
• Euler's identity e^{iπ} + 1 = 0 follows directly from the definition.

Worked Example

Compute sin(i) = (e^{−1} − e^{1})/(2i). Multiply by i/i: = −i(e − e^{−1})/2 = −i sinh(1) ≈ −1.1752i. And cos(i) = (e^{−1} + e^{1})/2 = cosh(1) ≈ 1.5431. Verify sin²z + cos²z = 1 at z = i: sin²(i) = (−1.1752i)² = −1.3811, cos²(i) = 1.5431² = 2.3811, sum = 1.0000. The identity survives even though |sin(i)| > 1. Logarithm check: Log(i) = ln 1 + iπ/2 = iπ/2, and e^{iπ/2} = i, confirming the inverse.

Common Misconceptions

• "e^z = 0 is solvable." Never: |e^z| = e^x > 0, so the exponential has no zeros.
• "log(ab) = log a + log b always." Principal branches agree only up to 2πi; cuts break naive equality.
• "|sin z| ≤ 1." True on the real axis only; on the imaginary axis sine grows like hyperbolic sine.

Real-World Applications

• AC signals as Re(Ve^{iωt}); impedance arithmetic is complex exponential arithmetic.
• Quantum time evolution e^{−iHt/ħ} is a complex exponential of the Hamiltonian.
• e^z maps strips to sectors; the logarithm straightens curved boundaries in conformal mapping.

The exponential wraps the plane around the punctured plane, the logarithm unwraps it at the cost of a branch cut, and trig functions become exponentials, unbounded off the real axis yet obeying every familiar identity.