Using the ratio test it is possible to show that this power series has an infinite radius of convergence, and so defines e < sup > z </ sup > for all complex z.

Using the Keck telescopes and a data set of 128 quasars at redshifts 0. 5 < z < 3, Webb et al.

Using Log we obtain the principal value of z < sup > α </ sup >, i. e.,

Using this approach, for fixed z the Weierstrass functions become modular functions of τ.

Using capital from Intel, Hewlett-Packard, Microsoft, and other major investors they designed a line of Itanium-based computers and software to emulate z / Architecture machines so that they could run zSeries operating systems, with zSeries channels for attaching real IBM equipment as well as virtual simulators for most hardware to minimize the need for IBM ’ s peripheral equipment.

Using the voltage divider ( R1, R2 and R3 ) allows choice of the arbitrary output voltage between U < sub > z </ sub > and U < sub > in </ sub >.

Using this, the general expression of the coefficient of z < sup > k + r </ sup > is

Using this root, we set the coefficient of z < sup > k + r − 2 </ sup > to be zero ( for it to be a solution ), which gives us the recurrence

Using the above definition for z, the CDF can be found as follows:

Using the method of moments, take expectations conditional on z to find

Using trigonometry, we can now rotate v < sub > x </ sub > by θ around z to obtain v < sub > x rot </ sub >.

* Using an adjective to refer to an element of a person, rather than using a noun to define them, for example, "... makes her look slutty " instead of "... is a slut "

* Using a positive context (" Inspired by " instead of " ripped off of " or " plagiarized ", " streamlining the workforce " for " laying off / firing ( workers )").

Using synthetic DNA, instead of there being 4 < sup > 3 </ sup > codons, if there are n new bases there could be as many as n < sup > 3 </ sup > codons.

Using a scanning point of light instead of full sample illumination confocal microscopy gives slightly higher resolution, and significant improvements in optical sectioning.

Using the same printer as for the Penny Black, Brazil opted for an abstract design instead of a portrait of Emperor Pedro II, so that his image would be not be disfigured by a postmark.

Using λ instead of φ ( n ) allows more choices for d. λ can also be defined using the Carmichael function, λ ( n ).

# Using a mathematical expression, such as a polynomial or a trigonometric function, and a single point on the corresponding curve instead of storing or transmitting the entire graphic curve or a series of points on it.

Using crop rotation farmers can keep their fields under continuous production, instead of letting them lie fallow, as well as reducing the need for artificial fertilizers, both of which can be expensive.

Using high-k dielectrics instead of silicon dioxide that is the conventional gate dielectric allows similar device performance, but with a thicker gate insulator, thus avoiding this current.

Using multiple components with load balancing, instead of a single component, may increase reliability through redundancy.

* periphrasis: Using several words instead of few

Note: Using a stack instead of a queue would turn this algorithm into a depth-first search.

Using, for example, the " mean anomaly " instead of " mean anomaly at epoch " means that time must be specified as a " seventh " orbital element.

Using an emergency number recognized by a GSM phone like 112 instead of another emergency number may be advantageous, since GSM phones and networks give special priority to emergency calls.

Using the Luftwaffe's limited resources to attack civilians instead of airfields and radar proved a major mistake.

) Using the < gml: pos > element instead of the < gml: coordinates > element, the same point can be represented as follows:

Using VxD drivers instead of WDM drivers in Windows 9x resulted in advanced ACPI states like hibernation being unavailable.

Using this system, the navigator does not immediately update every chart in the portfolio when a new Notice to Mariners arrives, instead creating a card for every chart and noting the correction on this card.

Using an alkyne instead of an alkene yields an enol, which tautomerizes into a ketone.

Using an alcohol instead of water yields an ether.

Using " weakly inaccessible " instead of " inaccessible ", similar definitions can be made for " weakly α-inaccessible ", " weakly hyper-inaccessible ", and " weakly α-hyper-inaccessible ".

Using an acceptance letter from the University of Wisconsin – Madison to be discharged from the service, Grant decided to attend the University of Minnesota instead.

Using standard chips instead of custom bond-out versions makes the technology ubiquitous and low cost, and eliminates most differences between the development and runtime environments.

Using a variable ( instead of a literal ) takes advantage of a compiler's checking.

* Using a credit / debit card or cash instead of cheques.

Using launch systems presently available, we are developing satellites to scout the world's weather ; ;

Hence S breaks up into uncountably many orbits under G. Using the axiom of choice, we could pick a single point from each orbit, obtaining an uncountable subset X of S with the property that all of its translates by G are disjoint from X.

Using the identity, we have

Using Zorn's lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Because of this essential uniqueness, we often speak of the algebraic closure of K, rather than an algebraic closure of K.

Using the above theorem it is easy to see that the original Borsuk Ulam statement is correct since if we take a map f: S < sup > n </ sup > → ℝ < sup > n </ sup > that does not equalize on any antipodes then we can construct a map g: S < sup > n </ sup > → S < sup > n-1 </ sup > by the formula

Using the right-hand rule, we expect the curl to be into the page.

Using the product rule in reverse, we get:

Using now the power series definition from above we see that for real values of x

Using the composition of functions we can rewrite this statement as follows:

Using these two examples, we can make some useful observations that motivate the major ideas in lambda calculus.

Using the definition of free variables, we may now define a capture-avoiding substitution.

Using our senses of sight, hearing, touch, smell, and taste, we perceive colors, sounds, textures, etc.

* Using the above methods for simple polygons, we discover that particular regions within the polygon may have their area multiplied by a factor which we call the density of the region.

Using this information we can perform a series of exchanges on the input array that puts it in order, moving items only once.

Using =, we find that the period of precession is given by:

Using relativistic relations for energy and momentum, we have

Using the chosen basis, we may write

Using differential calculus, we can determine the limit, or the value that Δy / Δx approaches as Δy and Δx get closer to zero ; it follows that this limit is the exact slope of the tangent.

Using the previous example, we would get the following:

Using this, we can get the relation, again choosing the right sign:

Using this Carnapian approach, Maher identifies a proposition which we intuitively ( and correctly ) know to be false, but which we easily confuse with the paradoxical conclusion.

Using Le Chatelier's principle, we can predict that the amount of methanol will increase, decreasing the total change in CO.