Defining Exponential and Trigonometric Functions Using Differential Equations

@article{Diamond2014DefiningEA,
  title={Defining Exponential and Trigonometric Functions Using Differential Equations},
  author={Harvey Diamond},
  journal={Mathematics Magazine},
  year={2014},
  volume={87},
  pages={37 - 42}
}
  • H. Diamond
  • Published 1 February 2014
  • Mathematics, Philosophy
  • Mathematics Magazine
Summary This note addresses the question of how to rigorously define the functions exp(x), sin(x), and cos(x), and develop their properties directly from that definition. We take a differential equations approach, defining each function as the solution of an initial value problem. Assuming only the basic existence/uniqueness theorem for solutions of linear differential equations, we derive the standard properties and identities associated with these functions. Our target audience is… 

Defining trigonometric functions via complex sequences

In the literature we find several different ways of introducing elementary functions. For the exponential function, we mention the following ways of characterising the exponential function: (a) (b) ,

References

SHOWING 1-3 OF 3 REFERENCES

Calculus and Analytic Geometry

A leaner, crisper, more accessible edition (according to the preface), for the widening range of students who need knowledge of the basic concepts. No bibliography. Annotation copyright Book News,

and D

  • Penney, Calculus and Analytic Geometry, 3rd ed., Prentice-Hall, Upper Saddle River, NJ,
  • 1990

and S

  • Bell, Mathematical Analysis for Modeling, CRC Press, Boca Raton, FL,
  • 1998