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## Taylor’s Formula

### Short Calculus (2002-01-01): 195-210 , January 01, 2002

We finally come to the point where we develop a method which allows us to compute the values of the elementary functions like sine, exp, and log. The method is to approximate these functions by polynomials, with an error term which is easily estimated. This error term will be given by an integral, and our first task is to estimate integrals. We then go through the elementary functions systematically, and derive the approximating polynomials.

## Completions

### Algebraic Number Theory (1994-01-01) 110: 31-55 , January 01, 1994

This chapter introduces the completions of number fields under the p-adic topologies, and also the completions obtained by embedding the number field into the real or complex numbers.

## Back Matter - Cyclotomic Fields I and II

### Cyclotomic Fields I and II (1990-01-01): 121 , January 01, 1990

## Calculus of Residues

### Complex Analysis (1985-01-01) 103: 165-195 , January 01, 1985

We have established all the theorems needed to compute integrals of analytic functions in terms of their power series expansions. We first give the general statements covering this situation, and then apply them to examples.

## Academia, Journalism, and Politics

### Challenges (1998-01-01): 1-222 , January 01, 1998

For three decades I have been interested in the area where the academic world meets the world of journalism and the world of politics. On several occasions I have had the opportunity to study how political opinions are passed off as science or scholarship. Some people have said—or “charged”!—that I am “politically motivated.” Of course I am politically motivated! But in what sense? I define “politics” to mean in the broad sense how society is organized, how one deals with social organizations, our relationship to government, how we arrive at decisions affecting the country and the world, the way ideas and information are disseminated in the media, the role of education, the way ideas are taught in schools and the universities, how information is processed (by the press, by individuals, by the educational system, by the government, etc.). I understand politics in that broad sense, and in that sense I am politically motivated. But my concern for politics does not mean that I support some faction, or some wing over another wing, say the left wing over the right wing; or that I support some “ism” ideology such as socialism, communism, or capitalism. I totally reject such factionalism.

## Kummer Theory over Cyclotomic Z p -extensions

### Cyclotomic Fields (1978-01-01) 59: 148-165 , January 01, 1978

In the last chapter we studied the ideal class groups in a Z_{p}-extension of a number field. Here we shall consider especially the cyclotomic Z_{p}-extension, and then Kummer extensions above it, as in Iwasawa [Iw 12], obtained by adjoining *p*^{n}th roots of units, *p*-units, and ideal classes of *p*-power order.

## The group of automorphisms of the modular function field

### Inventiones mathematicae (1971-09-01) 14: 253-254 , September 01, 1971

## Integration of Differential Forms

### Differential and Riemannian Manifolds (1995-01-01) 160: 284-306 , January 01, 1995

The material of this chapter is also contained in my book on real analysis [La 93], but it may be useful to the reader to have it also here in a rather self contained way, based only on standard properties of integration in Euclidean space.