Below you will find example sentences with "complex plane". The examples show how this phrase is used in natural context and which words often surround it.
Complex Plane in a sentence
Corpus data
- Displayed example sentences: 20
- Discovered as a combination around: complex
- Corpus frequency in the collocation scan: 12
- Phrase length: 2 words
- Average sentence length: 27.2 words
Sentence profile
- Phrase position: 1 start, 17 middle, 2 end
- Sentence types: 20 statements, 0 questions, 0 exclamations
Corpus analysis
- The phrase "complex plane" has 2 words and usually appears in the middle in these examples. The average sentence has 27.2 words and is mostly made up of statements.
- Around this phrase, patterns and context words such as example main complex plane exponential function, in the complex plane, function, roots and values stand out.
- In the phrase index, this combination connects with projective plane, apartment complex and complex numbers, linking the page to nearby combinations.
Example types with complex plane
This selection groups the examples by length and sentence type, making usage of the full phrase easier to scan:
For example: : main Complex plane Exponential function on the complex plane. (11 words)
In 1815 Poisson studied integrations along paths in the complex plane. (11 words)
As in the real case, the exponential function can be defined on the complex plane in several equivalent forms. (19 words)
These equations are in fact even valid for complex values of x, because both sides are entire (that is, holomorphic on the whole complex plane ) functions of x, and two such functions that coincide on the real axis necessarily coincide everywhere. (41 words)
Properties Square roots The two square roots of i in the complex plane The three cube roots of i in the complex plane i has two square roots, just like all complex numbers (except zero, which has a double root). (40 words)
The roots of the theory go back to the result of Émile Picard in 1879, showing that a non-constant complex-valued function which is analytic in the entire complex plane assumes all complex values save at most one. (39 words)
Example sentences (20)
Properties Square roots The two square roots of i in the complex plane The three cube roots of i in the complex plane i has two square roots, just like all complex numbers (except zero, which has a double root).
For example: : main Complex plane Exponential function on the complex plane.
Like the complex plane and split-complex number plane, the dual numbers are one of the realizations of planar algebra.
The roots of the theory go back to the result of Émile Picard in 1879, showing that a non-constant complex-valued function which is analytic in the entire complex plane assumes all complex values save at most one.
Klein showed that the modular group moves the fundamental region of the complex plane so as to tessellate that plane.
Complex graphs In the following graphs, the domain is the complex plane pictured, and the range values are indicated at each point by color.
Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part.
Practically speaking, stability requires that the transfer function complex poles reside * in the open left half of the complex plane for continuous time, when the Laplace transform is used to obtain the transfer function.
The idea of a complex number as a point in the complex plane (above) was first described by Caspar Wessel in 1799, although it had been anticipated as early as 1685 in Wallis's De Algebra tractatus.
These equations are in fact even valid for complex values of x, because both sides are entire (that is, holomorphic on the whole complex plane ) functions of x, and two such functions that coincide on the real axis necessarily coincide everywhere.
Topological groups The unit circle in the complex plane under complex multiplication is a Lie group and, therefore, a topological group.
A function that is equal to its Taylor series in an open interval (or a disc in the complex plane ) is known as an analytic function in that interval.
As a consequence of Liouville's theorem, any function that is entire on the whole Riemann sphere (complex plane and the point at infinity) is constant.
As in the real case, the exponential function can be defined on the complex plane in several equivalent forms.
Consider the resolvent function : which is a meromorphic function on the complex plane with values in the vector space of matrices.
Each zero has a basin of attraction in the complex plane, the set of all starting values that cause the method to converge to that particular zero.
Hasse's conjecture affirms that the L-function admits an analytic continuation to the whole complex plane and satisfies a functional equation relating, for any s, L(E, s) to L(E, 2 − s).
Hence f(z) takes on the value of every number in the complex plane except for zero infinitely often.
If f(x) is given by a convergent power series in an open disc (or interval in the real line) centered at b in the complex plane, it is said to be analytic in this disc.
In 1815 Poisson studied integrations along paths in the complex plane.