3.4: Inverse trigonometric/hyperbolic functions 3.5: Exponential and power functions 3.6: Branch points, branch cuts wi4243AP/wi4244AP: Complex Analysis week 3, Friday K. P. Hart Faculty EEMCS TU Delft Delft, 19 september, 2014 K. P. Hart wi4243AP/wi4244AP: Complex Analysis
3.4: Inverse trigonometric/hyperbolic functions 3.5: Exponential and power functions 3.6: Branch points, branch cuts Outline 3.4: Inverse trigonometric/hyperbolic functions 1 3.5: Exponential and power functions 2 Exponential functions Power functions 3.6: Branch points, branch cuts 3 Branch points Branch cuts An example K. P. Hart wi4243AP/wi4244AP: Complex Analysis
3.4: Inverse trigonometric/hyperbolic functions 3.5: Exponential and power functions 3.6: Branch points, branch cuts The arctangent function Wat does the complex arctan look like? Remember: tan is periodic, with period π . So if z = tan w , then z = tan( w + π ), z = tan( w − π ), . . . Hence arctan z has many values: w , w + π , w − π , . . . K. P. Hart wi4243AP/wi4244AP: Complex Analysis
3.4: Inverse trigonometric/hyperbolic functions 3.5: Exponential and power functions 3.6: Branch points, branch cuts The arctangent function Let’s solve z = tan w for w : Start with e iw − e − iw z = sin w cos w = 1 e iw + e − iw i and multiply numerator and denominator by e iw : e 2 iw − 1 e 2 iw + 1 or iz = e 2 iw − 1 z = 1 e 2 iw + 1 i Solve for e 2 iw : e 2 iw = 1 + iz 1 − iz Thus we find . . . K. P. Hart wi4243AP/wi4244AP: Complex Analysis
3.4: Inverse trigonometric/hyperbolic functions 3.5: Exponential and power functions 3.6: Branch points, branch cuts The arctangent function . . . upon taking the logarithm � 1 + iz � 2 iw = log 1 − iz and so � 1 + iz � � 1 + iz � w = 1 = 1 2 i log 2 i Log + k π 1 − iz 1 − iz And yes, the values differ by integer multiples of π K. P. Hart wi4243AP/wi4244AP: Complex Analysis
3.4: Inverse trigonometric/hyperbolic functions Exponential functions 3.5: Exponential and power functions Power functions 3.6: Branch points, branch cuts a z We know e z . What about 2 z ? Or even: 1 z ? In the R eal world: a x = e x ln a . In the C omplex world: a z = e z log a . Many functions: one for each value of log a . K. P. Hart wi4243AP/wi4244AP: Complex Analysis
3.4: Inverse trigonometric/hyperbolic functions Exponential functions 3.5: Exponential and power functions Power functions 3.6: Branch points, branch cuts 2 z So 2 z can be e z ln 2 e z ln 2+2 π iz e z ln 2 − 2 π iz . . . K. P. Hart wi4243AP/wi4244AP: Complex Analysis
3.4: Inverse trigonometric/hyperbolic functions Exponential functions 3.5: Exponential and power functions Power functions 3.6: Branch points, branch cuts 1 z And 1 z can be e z ln 1 = e 0 = 1 (constant) e 2 π iz (not constant) e − 2 π iz (also not constant) . . . K. P. Hart wi4243AP/wi4244AP: Complex Analysis
3.4: Inverse trigonometric/hyperbolic functions Exponential functions 3.5: Exponential and power functions Power functions 3.6: Branch points, branch cuts z a , integer a Integer a : unambiguously defined. If a > 0: z a = z × z × · · · × z � �� � a times If a < 0: 1 z a = z − a and, of course, z 0 = 1 K. P. Hart wi4243AP/wi4244AP: Complex Analysis
3.4: Inverse trigonometric/hyperbolic functions Exponential functions 3.5: Exponential and power functions Power functions 3.6: Branch points, branch cuts z a , rational a Rational a : many-valued (but finitely many): Arg z i × e 1 1 2 k π n i n = | z | n × e z n one value for each of k = 0, 1, . . . , n − 1. If a = m n , with gcd( m , n ) = 1, then z a = ( z 1 n ) m again n different values. K. P. Hart wi4243AP/wi4244AP: Complex Analysis
3.4: Inverse trigonometric/hyperbolic functions Exponential functions 3.5: Exponential and power functions Power functions 3.6: Branch points, branch cuts z a , real a In the real case we define z a = e a ln z for positive z and arbitrary a . If a is real and z is complex we use the complex logarithm: z a = e a log z = e a ln | z | + ia arg z = | z | a · e ia arg z Note: | z a | = | z | a , so “taking powers of moduli” still works. Similarly: arg z a = a · arg z , so “taking multiples of angles” still works. We actually have | z | a · e ia (Arg z +2 k π ) , infinitely many values. K. P. Hart wi4243AP/wi4244AP: Complex Analysis
3.4: Inverse trigonometric/hyperbolic functions Exponential functions 3.5: Exponential and power functions Power functions 3.6: Branch points, branch cuts z a , real a These functions are useful if you want to smooth out corners in a domain: z π z The angle on the left is 1 radian. K. P. Hart wi4243AP/wi4244AP: Complex Analysis
3.4: Inverse trigonometric/hyperbolic functions Exponential functions 3.5: Exponential and power functions Power functions 3.6: Branch points, branch cuts z a , arbitrary a Other a = α + i β : many-valued (infinitely many): z a = e a log z = e ( α + i β )(ln | z | + i arg z ) = e α ln | z |− β arg z · e i ( α arg z + β ln | z | ) = e α ln | z |− β (Arg z +2 k π ) · e i ( α (Arg z +2 k π )+ β ln | z | ) one value for each k ∈ Z . Note the β ln | z | ; this will turn a straight line through the origin into a spiral: if z = re i θ , with θ fixed, then z a = r α · e − βθ · e i αθ · e i β ln r . K. P. Hart wi4243AP/wi4244AP: Complex Analysis
3.4: Inverse trigonometric/hyperbolic functions Exponential functions 3.5: Exponential and power functions Power functions 3.6: Branch points, branch cuts i i Famous example (Euler). Note log i = 1 2 π i + 2 k π i so i i = e i log i = e i ( 1 2 π i +2 k π i ) = e − 1 2 π − 2 k π one (real) value for each integer k K. P. Hart wi4243AP/wi4244AP: Complex Analysis
3.4: Inverse trigonometric/hyperbolic functions Branch points 3.5: Exponential and power functions Branch cuts 3.6: Branch points, branch cuts An example √ z The function √ z is two-valued: � � � 2 i Arg z and 2 i Arg z + i π = − 1 1 1 2 i Arg z | z | e | z | e | z | e Choose a value for √− 1, say − √ 1 2 π i = − i (the second choice). 1 e Walk clockwise along the circle given by | z | = 1, retaining this choice. K. P. Hart wi4243AP/wi4244AP: Complex Analysis
3.4: Inverse trigonometric/hyperbolic functions Branch points 3.5: Exponential and power functions Branch cuts 3.6: Branch points, branch cuts An example √ z √− 1? i � � − 4 5 − 3 5 i − 3 5 − 4 5 i − 3 5 + 4 √ 3 5 + 4 5 i 5 i − i − 4 5 + 3 4 5 + 3 5 i 5 i � 5 − 4 3 � 5 i 4 5 − 3 5 i √ − 1 − 1 1 1 � 4 5 + 3 � 5 i − 3 5 + 4 5 i − 4 5 − 3 √ 5 − 3 4 5 i 5 i i � − 3 5 − 4 3 5 − 4 � 5 i 5 i − 3 5 + 4 − 4 5 + 3 5 i 5 i √− 1 − i K. P. Hart wi4243AP/wi4244AP: Complex Analysis
3.4: Inverse trigonometric/hyperbolic functions Branch points 3.5: Exponential and power functions Branch cuts 3.6: Branch points, branch cuts An example √ z After going round the circle once we end up at the other square root of − 1. We have moved to an other branch of √ z . We say 0 is a branch point of √ z . Also ∞ is a branch point of √ z (every circle around 0 is also a circle around ∞ ). K. P. Hart wi4243AP/wi4244AP: Complex Analysis
3.4: Inverse trigonometric/hyperbolic functions Branch points 3.5: Exponential and power functions Branch cuts 3.6: Branch points, branch cuts An example √ z , continue the walk √− 1 i � � − 4 5 + 3 5 i − 3 5 + 4 5 i − 3 5 + 4 5 + 4 3 √ 5 i 5 i i − 4 5 + 3 4 5 + 3 5 i 5 i � − 3 5 + 4 � 5 i 5 + 3 4 5 i √ − 1 − 1 1 and 1 � 4 5 − 3 � 5 i 3 5 − 4 5 i − 4 5 − 3 √ 5 − 3 4 5 i 5 i − i − 3 5 − 4 � 3 5 − 4 5 i � 5 i − 3 5 − 4 − 4 5 − 3 5 i 5 i √− 1 − i K. P. Hart wi4243AP/wi4244AP: Complex Analysis
3.4: Inverse trigonometric/hyperbolic functions Branch points 3.5: Exponential and power functions Branch cuts 3.6: Branch points, branch cuts An example √ z , continue the walk After going round the circle once more we end up at our original square root of − 1. We are back on our original branch of √ z . This makes 0 is a branch point of order 1 of √ z . Order: the number of times to go round a branch point to get back to the original value minus one. K. P. Hart wi4243AP/wi4244AP: Complex Analysis
3.4: Inverse trigonometric/hyperbolic functions Branch points 3.5: Exponential and power functions Branch cuts 3.6: Branch points, branch cuts An example √ z One way of splitting a many-valued function into usable single-valued branches is by cutting the plane along a suitable curve. For √ z a popular choice is the negative real axis, it connects the branch points 0 and ∞ . On the complement both branches are single-valued analytic functions. K. P. Hart wi4243AP/wi4244AP: Complex Analysis
3.4: Inverse trigonometric/hyperbolic functions Branch points 3.5: Exponential and power functions Branch cuts 3.6: Branch points, branch cuts An example log z log z has the same branch points, 0 and ∞ , as √ z : after going round a circle around 0 the argument of z has changed by 2 π (positive or negative) hence log z has changed by 2 π i . If you keep going (in the same direction) you will never get back to the original value; this branch point has order ∞ . Any line from 0 to ∞ can serve as a branch cut of log z ; usually one takes the negative real axis. K. P. Hart wi4243AP/wi4244AP: Complex Analysis
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