Ch.1:Complex Numbers Ch.1:Complex Numbers L1.3 Vectors and Polar Forms L1.3 Vectors and Polar Forms Vector Addition and Subtraction (cont.) Polar Forms of Complex Numbers (x,)=→(r,) =rcoso,y=rsin Parallelogram Law for addition of two vectors (or complex z=r(cos0+isin)=rcis0 numbers) z=x+iy r=la=Vr2+y2 Triangle Inequality:|z1+z<z+z2 The length of any side of a triangle is no greater than the sum 0= arctan(y/r),x >0; of the lengths of the other two sides arctan(y/x)±元，x<0 I to IV quadrant Corollary:z2l≤lal+l2-zl=→lz2l-lz1l≤lz2-zl argz=Argz+2kπ The difference of the lengths of any two sides of a triangle is 0 (k=0,±1，士2，.) no greater than the length of the third side Argz is the principal value of arg z Polar Coordinates Q:How to represent a product of two complex numbers in a 2-D plane? 4日10。+之+1生+意QG Ch.1:Complex Numbers Ch.1:Complex Numbers L13 Vectors and Polar Forms 14 The Complex Exponential Another Two Examples Euler's Equation The real exponential function f(x)=e where x is a real number 2122个 By replacing with z =z+iy,we get the complex Z2 exponential function f(z)=e= First,we postulate that the multiplication property should 0 persist:exti=eeiv,where e is still a real number and the second part eu needs to be defined According to Taylor'series expansion,we get the following 0 equation Geometric interpretation of the product Conjugate and the reciprocal ++++苦+… e=1+y+ (1)Ch.1: Complex Numbers 1.3 Vectors and Polar Forms Vector Addition and Subtraction (cont.) Parallelogram Law for addition of two vectors (or complex numbers) Triangle Inequality: |z1 + z2|≤|z1| + |z2| The length of any side of a triangle is no greater than the sum of the lengths of the other two sides Corollary:|z2|≤|z1| + |z2 − z1| =⇒ |z2|−|z1|≤|z2 − z1| The difference of the lengths of any two sides of a triangle is no greater than the length of the third side Ch.1: Complex Numbers 1.3 Vectors and Polar Forms Polar Forms of Complex Numbers (x, y) =⇒ (r, θ) x = r cos θ, y = r sin θ z = r(cos θ + isin θ) = rcisθ r = |z| = x2 + y2 θ =  arctan(y/x), x> 0; arctan(y/x) ± π, x < 0 I to IV quadrant arg z = Argz + 2kπ (k = 0, ±1, ±2,...) Argz is the principal value of arg z Q: How to represent a product of two complex numbers in a 2-D plane? Ch.1: Complex Numbers 1.3 Vectors and Polar Forms Another Two Examples Ch.1: Complex Numbers 1.4 The Complex Exponential Euler’s Equation The real exponential function f(x) = ex where x is a real number By replacing x with z = x + iy, we get the complex exponential function f(z) = ez First, we postulate that the multiplication property should persist: ex+iy = exeiy, where ex is still a real number and the second part eiy needs to be defined According to Taylor’ series expansion, we get the following equation ey =1+ y + y22! + y33! + y44! + y55! + ... (1)