representation; for instance, Euler's formula follows from Taylor series expansions for trigonometric and exponential functions. This result is of fundamental importance in such fields as harmonic analysis. Approximations using the first few terms of a Taylor series can make otherwise unsolvable problems possible for a restricted domain; this approach is often used in physics. Approximation error and convergence Pictured is an accurate approximation of around the point . The pink curve is a polynomial of degree seven: The error in this approximation is no more than . For a full cycle centered at the origin () the error is less than 0.08215. In particular, for , the error is less than 0.000003. In contrast, also shown is a picture of the natural logarithm function and some of its Taylor polynomials around . These approximations converge to the function only in the region ; outside of this region the higher-degree Taylor polynomials are worse approximations for the function. The error incurred in approximating a function by its th-degree Taylor polynomial is called the remainder or residual and is denoted by the function . Taylor's theorem can be used to obtain a bound on the size of the remainder. In general, Taylor series need not be convergent at all. And in fact the set of functions with a convergent Taylor series is a meager set in the Fréchet space of smooth functions. And even if the Taylor series of a function does converge, its limit need not in general be equal to the value of the function . For example, the function is infinitely differentiable at , and has all derivatives zero there. Consequently, the Taylor series of about is identically zero. However, is not the zero function, so does not equal its Taylor series around the origin. Thus, is an example of a non-analytic smooth function. In real analysis, this example shows that there are infinitely differentiable functions whose Taylor series are not equal to even if they converge. By contrast, the holomorphic functions studied in complex analysis always possess a convergent Taylor series, and even the Taylor series of meromorphic functions, which might have singularities, never converge to a value different from the function itself. The complex function , however, does not approach 0 when approaches 0 along the imaginary axis, so it is not continuous in the complex plane and its Taylor series is undefined at 0. More generally,