About: Proof of Euler's formula   Sponge Permalink

An Entity of Type : owl:Thing, within Data Space : 134.155.108.49:8890 associated with source dataset(s)

Here is a proof of Euler's formula using Taylor series expansions as well as basic facts about the powers of i: and so on. The functions ex, cos(x) and sin(x) (assuming x is real) can be written as: and for complex z we define each of these function by the above series, replacing x with iz. This is possible because the radius of convergence of each series is infinite. We then find that The rearrangement of terms is justified because each series is absolutely convergent. Taking z = x to be a real number gives the original identity as Euler discovered it.

AttributesValues
rdfs:label
  • Proof of Euler's formula
rdfs:comment
  • Here is a proof of Euler's formula using Taylor series expansions as well as basic facts about the powers of i: and so on. The functions ex, cos(x) and sin(x) (assuming x is real) can be written as: and for complex z we define each of these function by the above series, replacing x with iz. This is possible because the radius of convergence of each series is infinite. We then find that The rearrangement of terms is justified because each series is absolutely convergent. Taking z = x to be a real number gives the original identity as Euler discovered it.
dcterms:subject
abstract
  • Here is a proof of Euler's formula using Taylor series expansions as well as basic facts about the powers of i: and so on. The functions ex, cos(x) and sin(x) (assuming x is real) can be written as: and for complex z we define each of these function by the above series, replacing x with iz. This is possible because the radius of convergence of each series is infinite. We then find that The rearrangement of terms is justified because each series is absolutely convergent. Taking z = x to be a real number gives the original identity as Euler discovered it.
Alternative Linked Data Views: ODE     Raw Data in: CXML | CSV | RDF ( N-Triples N3/Turtle JSON XML ) | OData ( Atom JSON ) | Microdata ( JSON HTML) | JSON-LD    About   
This material is Open Knowledge   W3C Semantic Web Technology [RDF Data] Valid XHTML + RDFa
OpenLink Virtuoso version 07.20.3217, on Linux (x86_64-pc-linux-gnu), Standard Edition
Data on this page belongs to its respective rights holders.
Virtuoso Faceted Browser Copyright © 2009-2012 OpenLink Software