[Music] [Music] [Music] um foreign foreign v upper real part in the dou u by dou x with respect to x in differentiating dou u by dou y underground with respect to y k partial differentiator imaginary part with respect to x with respect to y k differentiator x is equal to dou v by dou y real partner with respect to x and a partial differentiating in the imaginary part with respect to y k partial differentiating the same idea at the polar dou u by dou y equal to minus dou v by dou x is okay proof x plus delta x plus i into x plus delta x real pattern plus i into imaginary partnering y plus is it plus delta set minus f4 facet by delta is it the limit is it tends to zero f of z plus delta is at minus f of set by delta is the reasonable derivative limit x tends to zero f of x plus delta x minus f of x by delta x same way in our formula y plus is delta x comma y plus delta y plus i into v of x plus delta x comma y plus delta y okay okay zero f4 is said plus delta i said minus f4 is set by delta is it the number of formulas x plus delta x comma y plus delta y plus i into v of x plus delta x comma y plus delta y substituted u of x comma y plus i into v of x comma y and dump the function the whole by delta i said i put a whole by delta e sit in the vertical x plus delta x plus i into y plus oh sorry delta is it delta x plus i delta y if i substitute you right u of x plus delta x comma y plus delta y minus sin david u of x comma y the whole by the whole by delta x plus i delta y okay so minus v of x comma y the whole by delta x plus i delta delta x tends to 0 by delta y equal to zero delta y equal to zero over dna delta x tends to zero either delta y equal to zero okay f dash of is it is equal minus of u of x comma y the whole by delta x and that delta y is that the delta we know that plus i into limit delta x tends to zero delta x comma y minus v of x comma y the whole by delta x minus u of x comma y the whole by delta x with respect to x in the u in a differentiated dou u by dou x so this is equal to dou u by dou x plus i into okay [Applause] 000 delta x is equal to 0 for k is 2 to say path delta y tends to 0 as delta x is equal to zero delta x is equal to zero on the other corner up underneath f dash of the name f dash officer is equal to limit delta x tends to zero delta y tends to zero delta y matrix u of x plus delta x is zero and u of x comma y plus delta y minus u of x comma y by inventory and the delta x plus i delta y l delta x limit delta y tends to 0 v of x comma y plus delta y minus v of x comma y by delta x plus i delta value delta x is equal to minus i into uy plus b y and then in bracket equation number two number number f dash of facet is equal to minus i into uy plus v y of the number of equations f dash of is it is equal to u x plus i v x negative number left natural facet is equal to infinity minus i u y plus v a negative another u x plus i v x is equal to equation two naught another minus i u y plus v y in another video okay even the real part equating real particular real patterning is minus vx find its derivative current so that f4 facet is analytic and also find it and also find its minus defined and continuous in some neighborhood of a point he said equal to x plus iy and differentiability said itself cauchy riemann equation the either dou u by dou x dou u by dou y dou v by dou x dou v by dou y exist and satisfy squashy riemann equation okay uh our equation 2018.99 the following function satisfy cr equation cr equation one and then in cauchy riemann equation and also find its derivative derivative x square plus 2 i x y minus i i y square nand plus i y the whole square another remainder minus y squared so other than minus y squared i think minus either i squared a minus one another minus y square right so it's gonna make a real pattern on this x square minus y square and then plus two in x square minus y square and imaginary part b is equal to two x minus okay i'm going to mark it [Applause] [Applause] with respect to y k differentiating y interval t 1 2 x u x is equal to b u x is equal to v and u y is equal to minus v x and negative you y known a minus two over n but number of v x naught minus sign in the lab and equation is satisfied therefore cr equation therefore cr equation is satisfied see our equation satisfaction therefore for wizard is analytic therefore f is it is analytic y plus i sine y negative okay that is equal to e raised to x into cos y plus i into e raised to x into sine y naught for real part y imaginary part y therefore real part u is equal to e raise to x into cos y and imaginary part v is equal to e raise to x into sine y okay partial derivative dou u by dou x known minus sine y okay x is equal to e raised to x derivative e raised to x nano sine y over the constant at v y naught and on a double v by dou by n with respect to y k differentiating is satisfied therefore cr equation is satisfied see our equation satisfaction otherwise it plus i sine y cos theta plus i sine theta and e raised to i theta either cos y plus i sine minus e raised to i y theta but out of the step x raised to m into x raised to n equal to x raised to m plus negative t raised to e raised to 2 into x minus i y inverted that is e raised to 2 x into e raised to minus 2 i y number nine i'll just split the e raised to 2 x into cos 2 y minus i sine 2 y that is e raised to 2 x into goes to y plus i into and sorry minus i into e raised to 2 x into sine y and squared function on the previous okay [Music]