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ди ди de 40. Project Problem: Cauchy-Riemann equations in polar form. In this problem we express the Cauchy-Riemann equations in polar coordinates. Recall the relationships between Cartesian and polar coordinates: x = rcos 0 and y = rsin 0. For convenience, we denote by 4, the derivatives of u(rcos 0, rsin 0) = u(x, y) with respect to r and 0, respectively. (a) The multivariable chain rule from calculus [see (2.4.13)] states that ди дх ди ду + дх доду до Jy dx ду ду ди ди дх ди ду ди + = Jr dx dr dy dre αν ду дх ду ду ду = + + др дх ду ду др’дө дх до ду до on some open set on which these derivatives exist. Show that ди ди ди ди ди ди = cos 0 + sin e = -rsin 0 +rcose dr дх ду до дх ду αν αν av av αν av = cos 0 + sin e = -rsin +rcos e ar дх ду до дх ду (b) Derive the polar form of the Cauchy-Riemann equations: ди Ju dr re 1 av до αν 1 ди and dr r de (2.5.14) Thus we can state Theorem 2.5.1 in polar form as follows: The function f = u+iv, u, v real-valued, is analytic on an open subset U of C if and only if u, v are differentiable on U and satisfy (2.5.14). (c) Show that if f = u+iv is analytic on U, then for rele EU we have f' ƒ'(rel®) = e−10 (344 αν +i (2.5.15) In Exercises 41 and 42 use the polar form (2.5.14) of the Cauchy-Riemann equations to verify the analyticity and evaluate the derivatives of the functions. 41. (cos(ne) + i sin(ne)) (n= ±1, ±2,...). == 42. Logz In|z|+i Arg z, z is not a negative real number nor zero.

ди ди de 40. Project Problem: Cauchy-Riemann equations in polar form. In this problem we express the Cauchy-Riemann equations in polar coordinates. Recall the relationships between Cartesian and polar coordinates: x = rcos 0 and y = rsin 0. For convenience, we denote by 4, the derivatives of u(rcos 0, rsin 0) = u(x, y) with respect to r and 0, respectively. (a) The multivariable chain rule from calculus [see (2.4.13)] states that ди дх ди ду + дх доду до Jy dx ду ду ди ди дх ди ду ди + = Jr dx dr dy dre αν ду дх ду ду ду = + + др дх ду ду др’дө дх до ду до on some open set on which these derivatives exist. Show that ди ди ди ди ди ди = cos 0 + sin e = -rsin 0 +rcose dr дх ду до дх ду αν αν av av αν av = cos 0 + sin e = -rsin +rcos e ar дх ду до дх ду (b) Derive the polar form of the Cauchy-Riemann equations: ди Ju dr re 1 av до αν 1 ди and dr r de (2.5.14) Thus we can state Theorem 2.5.1 in polar form as follows: The function f = u+iv, u, v real-valued, is analytic on an open subset U of C if and only if u, v are differentiable on U and satisfy (2.5.14). (c) Show that if f = u+iv is analytic on U, then for rele EU we have f' ƒ'(rel®) = e−10 (344 αν +i (2.5.15) In Exercises 41 and 42 use the polar form (2.5.14) of the Cauchy-Riemann equations to verify the analyticity and evaluate the derivatives of the functions. 41. (cos(ne) + i sin(ne)) (n= ±1, ±2,...). == 42. Logz In|z|+i Arg z, z is not a negative real number nor zero.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.5: Polar Coordinates
Problem 98E
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ди ди de 40. Project Problem: Cauchy-Riemann equations in polar form. In this problem we express the Cauchy-Riemann equations in polar coordinates. Recall the relationships between Cartesian and polar coordinates: x = rcos 0 and y = rsin 0. For convenience, we denote by 4, the derivatives of u(rcos 0, rsin 0) = u(x, y) with respect to r and 0, respectively. (a) The multivariable chain rule from calculus [see (2.4.13)] states that ди дх ди ду + дх доду до Jy dx ду ду ди ди дх ди ду ди + = Jr dx dr dy dre αν ду дх ду ду ду = + + др дх ду ду др’дө дх до ду до on some open set on which these derivatives exist. Show that ди ди ди ди ди ди = cos 0 + sin e = -rsin 0 +rcose dr дх ду до дх ду αν αν av av αν av = cos 0 + sin e = -rsin +rcos e ar дх ду до дх ду (b) Derive the polar form of the Cauchy-Riemann equations: ди Ju dr re 1 av до αν 1 ди and dr r de (2.5.14) Thus we can state Theorem 2.5.1 in polar form as follows: The function f = u+iv, u, v real-valued, is analytic on an open subset U of C if and only if u, v are differentiable on U and satisfy (2.5.14). (c) Show that if f = u+iv is analytic on U, then for rele EU we have f' ƒ'(rel®) = e−10 (344 αν +i (2.5.15) In Exercises 41 and 42 use the polar form (2.5.14) of the Cauchy-Riemann equations to verify the analyticity and evaluate the derivatives of the functions. 41. (cos(ne) + i sin(ne)) (n= ±1, ±2,...). == 42. Logz In|z|+i Arg z, z is not a negative real number nor zero.
Transcribed Image Text:ди ди de 40. Project Problem: Cauchy-Riemann equations in polar form. In this problem we express the Cauchy-Riemann equations in polar coordinates. Recall the relationships between Cartesian and polar coordinates: x = rcos 0 and y = rsin 0. For convenience, we denote by 4, the derivatives of u(rcos 0, rsin 0) = u(x, y) with respect to r and 0, respectively. (a) The multivariable chain rule from calculus [see (2.4.13)] states that ди дх ди ду + дх доду до Jy dx ду ду ди ди дх ди ду ди + = Jr dx dr dy dre αν ду дх ду ду ду = + + др дх ду ду др’дө дх до ду до on some open set on which these derivatives exist. Show that ди ди ди ди ди ди = cos 0 + sin e = -rsin 0 +rcose dr дх ду до дх ду αν αν av av αν av = cos 0 + sin e = -rsin +rcos e ar дх ду до дх ду (b) Derive the polar form of the Cauchy-Riemann equations: ди Ju dr re 1 av до αν 1 ди and dr r de (2.5.14) Thus we can state Theorem 2.5.1 in polar form as follows: The function f = u+iv, u, v real-valued, is analytic on an open subset U of C if and only if u, v are differentiable on U and satisfy (2.5.14). (c) Show that if f = u+iv is analytic on U, then for rele EU we have f' ƒ'(rel®) = e−10 (344 αν +i (2.5.15) In Exercises 41 and 42 use the polar form (2.5.14) of the Cauchy-Riemann equations to verify the analyticity and evaluate the derivatives of the functions. 41. (cos(ne) + i sin(ne)) (n= ±1, ±2,...). == 42. Logz In|z|+i Arg z, z is not a negative real number nor zero.
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