Theory of Functions of a Complex Variable
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GOURSAT-HEDRICK, Mathematical Analysis, Vol. I, p. 19. S. E. R.
34- CONFORMAL REPRESENTATION 183 the equation becomes infinitesimal with s z\ and z 3 z. If e is omitted in this equation, we obtain (except for the symbols) equation (17) of 10, whose geometrical significance was dis cussed there. We thus have an answer to the proposed ques tion ; it can be formulated as follows : I. If w is a regular function of z, then every triangle of the z-p!ane whose sides are infinitesimals of the same orde...r, is similar to the corresponding triangle of the w-plane up to infinitesimals of higher order, that is, ratio of sides and angles of the one differ only by infinitesimals from the corresponding parts of the other.
In particular, if we apply the results of 10 for the finite triangles discussed there to the infinitesimal triangles just men tioned, it follows that : II. The absolute value of the derivative - - at a point of the dz z-plane gives the scale of similarity* at that poi tit, that is, gives the factor by which the length of an infinitesimal arc of the z-plane must be multiplied in order to obtain the length of the corresponding arc of the w-planc.
What to read after Theory of Functions of a Complex Variable?
You can find similar books in the "Read Also" column, or choose other free books by Heinrich Burkhardt , Heinrich Friedrich Karl Ludwig Burkhardt to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
GOURSAT-HEDRICK, Mathematical Analysis, Vol. I, p. 19. S. E. R.
34- CONFORMAL REPRESENTATION 183 the equation becomes infinitesimal with s z\ and z 3 z. If e is omitted in this equation, we obtain (except for the symbols) equation (17) of 10, whose geometrical significance was dis cussed there. We thus have an answer to the proposed ques tion ; it can be formulated as follows : I. If w is a regular function of z, then every triangle of the z-p!ane whose sides are infinitesimals of the same orde...r, is similar to the corresponding triangle of the w-plane up to infinitesimals of higher order, that is, ratio of sides and angles of the one differ only by infinitesimals from the corresponding parts of the other.
In particular, if we apply the results of 10 for the finite triangles discussed there to the infinitesimal triangles just men tioned, it follows that : II. The absolute value of the derivative - - at a point of the dz z-plane gives the scale of similarity* at that poi tit, that is, gives the factor by which the length of an infinitesimal arc of the z-plane must be multiplied in order to obtain the length of the corresponding arc of the w-planc.
What to read after Theory of Functions of a Complex Variable?
You can find similar books in the "Read Also" column, or choose other free books by Heinrich Burkhardt , Heinrich Friedrich Karl Ludwig Burkhardt to read onlineMoreLess
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