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We now start to build up some examples of analytic functions. We have already seen that the function f(z) = z is entire. In this section we will generalise this to show that so is any polynomial P(z). We will also see that ratios of polynomials are also analytic everywhere but on a finite set of points in the complex plane where the denominator vanishes. There are many ways to do this, but one illuminating way is to show that complex linear combinations of analytic functions are analytic and that products of analytic functions are analytic functions. Let f(z) be an analytic function on some open subset U ⊂ C, and let α be a complex number. Then it is easy to see that the function α f(z) is also analytic on U. Indeed, from the definition (2.6) of the derivative, we see that

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分析関数の例をいくつか作り始めます。関数f（z）= zが完全であることはすでに見ました。この節では、これを一般化して任意の多項式P（z）もそうであることを示します。また、多項式の比率もどこでも解析的であることがわかりますが、分母が

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We will begin making some examples of analysis functions. We have already seen that the function f (z) = z is complete. In this section we generalize this to indicate that any polynomial P (z) is also. Also, it turns out that the ratio of polynomials is also analytical everywhere, but if the denominator is

INTO JAPANESE

分析関数の例をいくつか作り始めます。関数f（z）= zが完全であることはすでに見ました。この節では、これを一般化して任意の多項式P（z）もそうであることを示します。また、多項式の比率も至る所で解析的であることがわかりますが、分母が

BACK INTO ENGLISH

We will begin making some examples of analysis functions. We have already seen that the function f (z) = z is complete. In this section we generalize this to indicate that any polynomial P (z) is also. Also, it turns out that the ratio of polynomials is analytical everywhere, but if the denominator is

INTO JAPANESE

分析関数の例をいくつか作り始めます。関数f（z）= zが完全であることはすでに見ました。この節では、これを一般化して任意の多項式P（z）もそうであることを示します。また、多項式の比率は至る所で分析的ですが、分母が

BACK INTO ENGLISH

We will begin making some examples of analysis functions. We have already seen that the function f (z) = z is complete. In this section we generalize this to indicate that any polynomial P (z) is also. Also, the ratio of polynomials is analytical everywhere, but the denominator is

INTO JAPANESE

分析関数の例をいくつか作り始めます。関数f（z）= zが完全であることはすでに見ました。この節では、これを一般化して任意の多項式P（z）もそうであることを示します。また、多項式の比率は至る所で解析的ですが、分母は

BACK INTO ENGLISH

We will begin making some examples of analysis functions. We have already seen that the function f (z) = z is complete. In this section we generalize this to indicate that any polynomial P (z) is also. Also, the ratio of polynomials is analytical everywhere, but the denominator is