Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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Conjugation laws, , multiplicativity of modulus, and the triangle inequality
Statement
For complex , conjugation respects sums and products, , , and . The conventions and prerequisite facts used below are recorded in Real and imaginary parts, complex conjugation, and modulus, The complex numbers form a field, and every nonzero has inverse , The -norms for rational , and , Each is a norm on , and the induced metrics are exactly , and of the published metric-spaces page.
Facts & Assumptions
Given: and .
Proof
Expand the coordinate definitions to prove the conjugation laws and .
Squaring both nonnegative sides proves multiplicativity of the modulus.
The Euclidean norm triangle inequality on is exactly .
Depends on
- Real and imaginary parts, complex conjugation, and modulus
- The complex numbers form a field, and every nonzero $x+iy$ has inverse $(x-iy)/(x^2+y^2)$
- The $p$-norms $\lVert x\rVert_p$ for rational $p \ge 1$, and $\lVert x\rVert_\infty$
- Each $\lVert\cdot\rVert_p$ is a norm on $\mathbb{R}^n$, and the induced metrics are exactly $d_1$, $d_2$ and $d_\infty$ of the published metric-spaces page
Used by
- exp(x+iy)=eˣ(cos y+i sin y), |exp(x+iy)|=eˣ, and e^iπ+1=0 Corollary
- The complex geometric power series has radius 1 and sums to 1/(1-z) for |z|<1 Example
- A nonconstant complex polynomial tends to infinite modulus and attains a global minimum modulus Lemma
- A nonzero value of a nonconstant complex polynomial cannot be a local minimum of its modulus Lemma
- The Cauchy product of two absolutely convergent complex series converges absolutely to the product of their sums Lemma
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 107 results over 17 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- J. Lebl, Basic Analysis I: Complex Numbers and the Complex Exponential (standard reference, not scraped)