Complex identities — explanatory proofs

What this is?

A single-page explainer of classic complex-number identities. Each card states the identity, gives a short reason, and then performs a numeric check right in the browser to confirm the result. No external libraries or servers are used.

Numbers are shown to ~12 decimals.

sqrt(-1)

Answer: √(-1) = i

Reason: −1 = e^iπ (principal). √(e^iπ) = e^iπ/2 = i.

Check: i² =

exp(i·π) + 1

Answer: e^iπ + 1 = 0

Reason: Euler: e^iθ = cos θ + i sin θ ⇒ e^iπ = −1.

Check: e^iπ + 1 ≈

log(-1)

Answer: log(−1) = iπ (principal)

Reason: log z = ln|z| + i Arg(z); |−1| = 1 ⇒ ln|z| = 0, Arg(−1) = π.

Check: exp(log(−1)) =

log(-i)

Answer: log(−i) = −iπ/2 (principal)

Reason: |−i| = 1, Arg(−i) = −π/2 ⇒ log(−i) = 0 + i(−π/2).

Check: exp(log(−i)) =

log(i·x) with x = 0.73

Answer: log(i·x) = ln x + iπ/2 (x>0)

Reason: Arg(i x) = π/2 (principal), |i x| = x ⇒ ln|·| = ln x.

Check: exp(log(i·x)) ≈   and   i·x =

cos(i·x) with x = 0.73

Answer: cos(i x) = cosh x

Reason: cos z = (e^iz + e^−iz)/2; set z = i x ⇒ (e^−x + e^x)/2.

Check: cos(i x) ≈   ;   cosh(x) =

sin(i·x) with x = 0.73

Answer: sin(i x) = i·sinh x

Reason: sin z = (e^iz − e^−iz)/(2i); set z = i x ⇒ i(e^−x − e^x)/2.

Check: sin(i x) ≈   ;   i·sinh(x) =

asin(i·x) with x = 0.73

Answer: asin(i x) = i·asinh x

Reason: Definition asin z = −i·log( i z + √(1 − z²) ); put z = i x and use √(1 + x²) − x > 0; also ln(√(1 + x²) − x) = −ln(x + √(1 + x²)).

Check: i·asinh(x) =   ;   −i·ln(√(1 + x²) − x) =

sqrt(i)

Answer: √i = (1 + i)/√2

Reason: i = e^iπ/2; √i = e^iπ/4 = cos(π/4) + i sin(π/4) = (1 + i)/√2.

Check: (√i)² ≈

acos(2)

Answer: acos(2) = i·ln(2 + √3)

Reason: For |z| > 1, acos z = i ln(z + √(z² − 1)) (principal). For z = 2: √3 and ln(2 + √3).

Check: cos(acos(2)) = cos(i·ln(2 + √3)) = cosh(ln(2 + √3)) ≈

asin(2)

Answer: asin(2) = π/2 − i·ln(2 + √3)

Reason: asin z = π/2 − acos z and acos(2) = i ln(2 + √3).

Check: sin(asin(2)) = cosh(ln(2 + √3)) ≈

i^i

Answer: i^i = e^−π/2

Reason: i^i = e^{i log(i)} with principal log(i) = iπ/2 ⇒ exponent = −π/2 (real).

Check: numeric value ≈