wi4243AP/wi4244AP: Complex Analysis week 3, Friday K. P. Hart - - PowerPoint PPT Presentation

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wi4243AP/wi4244AP: Complex Analysis week 3, Friday K. P. Hart - - PowerPoint PPT Presentation

Jan 30, 2024 318 likes •575 views

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

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SLIDE 1

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

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SLIDE 2

3.4: Inverse trigonometric/hyperbolic functions 3.5: Exponential and power functions 3.6: Branch points, branch cuts

Outline

1

3.4: Inverse trigonometric/hyperbolic functions

2

3.5: Exponential and power functions Exponential functions Power functions

3

3.6: Branch points, branch cuts Branch points Branch cuts An example

  • K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

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SLIDE 3

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

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SLIDE 4

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 z = sin w cos w = 1 i eiw − e−iw eiw + e−iw and multiply numerator and denominator by eiw: z = 1 i e2iw − 1 e2iw + 1 or iz = e2iw − 1 e2iw + 1 Solve for e2iw: e2iw = 1 + iz 1 − iz Thus we find . . .

  • K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

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SLIDE 5

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 2iw = log 1 + iz 1 − iz

  • and so

w = 1 2i log 1 + iz 1 − iz

  • = 1

2i Log 1 + iz 1 − iz

  • + kπ

And yes, the values differ by integer multiples of π

  • K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

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SLIDE 6

3.4: Inverse trigonometric/hyperbolic functions 3.5: Exponential and power functions 3.6: Branch points, branch cuts Exponential functions Power functions

az

We know ez. What about 2z? Or even: 1z? In the Real world: ax = ex ln a. In the Complex world: az = ez log a. Many functions: one for each value of log a.

  • K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

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SLIDE 7

3.4: Inverse trigonometric/hyperbolic functions 3.5: Exponential and power functions 3.6: Branch points, branch cuts Exponential functions Power functions

2z

So 2z can be ez ln 2 ez ln 2+2πiz ez ln 2−2πiz . . .

  • K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

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SLIDE 8

3.4: Inverse trigonometric/hyperbolic functions 3.5: Exponential and power functions 3.6: Branch points, branch cuts Exponential functions Power functions

1z

And 1z can be ez ln 1 = e0 = 1 (constant) e2πiz (not constant) e−2πiz (also not constant) . . .

  • K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

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SLIDE 9

3.4: Inverse trigonometric/hyperbolic functions 3.5: Exponential and power functions 3.6: Branch points, branch cuts Exponential functions Power functions

za, integer a

Integer a: unambiguously defined. If a > 0: za = z × z × · · · × z

  • a times

If a < 0: za = 1 z−a and, of course, z0 = 1

  • K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

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SLIDE 10

3.4: Inverse trigonometric/hyperbolic functions 3.5: Exponential and power functions 3.6: Branch points, branch cuts Exponential functions Power functions

za, rational a

Rational a: many-valued (but finitely many): z

1 n = |z| 1 n × e Arg z n

i × e

2kπ n i

  • ne value for each of k = 0, 1, . . . , n − 1.

If a = m

n , with gcd(m, n) = 1, then

za = (z

1 n )m

again n different values.

  • K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

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SLIDE 11

3.4: Inverse trigonometric/hyperbolic functions 3.5: Exponential and power functions 3.6: Branch points, branch cuts Exponential functions Power functions

za, real a

In the real case we define za = ea ln z for positive z and arbitrary a. If a is real and z is complex we use the complex logarithm: za = ea log z = ea ln |z|+ia arg z = |z|a · eia arg z Note: |za| = |z|a, so “taking powers of moduli” still works. Similarly: arg za = a · arg z, so “taking multiples of angles” still works. We actually have |z|a · eia(Arg z+2kπ), infinitely many values.

  • K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

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SLIDE 12

3.4: Inverse trigonometric/hyperbolic functions 3.5: Exponential and power functions 3.6: Branch points, branch cuts Exponential functions Power functions

za, real a

These functions are useful if you want to smooth out corners in a domain: z zπ The angle on the left is 1 radian.

  • K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

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SLIDE 13

3.4: Inverse trigonometric/hyperbolic functions 3.5: Exponential and power functions 3.6: Branch points, branch cuts Exponential functions Power functions

za, arbitrary a

Other a = α + iβ: many-valued (infinitely many): za = ea log z = e(α+iβ)(ln |z|+i arg z) = eα ln |z|−β arg z · ei(α arg z+β ln |z|) = eα ln |z|−β(Arg z+2kπ) · ei(α(Arg z+2kπ)+β ln |z|)

  • ne value for each k ∈ Z.

Note the β ln |z|; this will turn a straight line through the origin into a spiral: if z = reiθ, with θ fixed, then za = rα · e−βθ · eiαθ · eiβ ln r.

  • K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

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SLIDE 14

3.4: Inverse trigonometric/hyperbolic functions 3.5: Exponential and power functions 3.6: Branch points, branch cuts Exponential functions Power functions

ii

Famous example (Euler). Note log i = 1 2πi + 2kπi so ii = ei log i = ei( 1

2 πi+2kπi) = e− 1 2 π−2kπ

  • ne (real) value for each integer k
  • K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

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SLIDE 15

3.4: Inverse trigonometric/hyperbolic functions 3.5: Exponential and power functions 3.6: Branch points, branch cuts Branch points Branch cuts An example

√z

The function √z is two-valued:

  • |z|e

1 2 i Arg z and

  • |z|e

1 2 i Arg z+iπ = −

  • |z|e

1 2 i Arg z

Choose a value for √−1, say − √ 1e

1 2 πi = −i (the second choice).

Walk clockwise along the circle given by |z| = 1, retaining this choice.

  • K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

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SLIDE 16

3.4: Inverse trigonometric/hyperbolic functions 3.5: Exponential and power functions 3.6: Branch points, branch cuts Branch points Branch cuts An example

√z

−1 √−1 − 4

5 + 3 5i

  • − 4

5 + 3 5i

− 3

5 + 4 5i

  • − 3

5 + 4 5i

i √ i

3 5 + 4 5i

  • − 3

5 + 4 5i 4 5 + 3 5i

  • 4

5 + 3 5i

1 √ 1

4 5 − 3 5i

  • 4

5 − 3 5i 3 5 − 4 5i

  • 3

5 − 4 5i

−i √ −i − 3

5 − 4 5i

  • − 3

5 − 4 5i

− 4

5 − 3 5i

  • − 4

5 − 3 5i

−1 √−1?

  • K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

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SLIDE 17

3.4: Inverse trigonometric/hyperbolic functions 3.5: Exponential and power functions 3.6: Branch points, branch cuts 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

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SLIDE 18

3.4: Inverse trigonometric/hyperbolic functions 3.5: Exponential and power functions 3.6: Branch points, branch cuts Branch points Branch cuts An example

√z, continue the walk

−1 √−1 − 4

5 + 3 5i

  • − 4

5 + 3 5i

− 3

5 + 4 5i

  • − 3

5 + 4 5i

i √ i

3 5 + 4 5i

  • − 3

5 + 4 5i 4 5 + 3 5i

  • 4

5 + 3 5i

1 and √ 1

4 5 − 3 5i

  • 4

5 − 3 5i 3 5 − 4 5i

  • 3

5 − 4 5i

−i √ −i − 3

5 − 4 5i

  • − 3

5 − 4 5i

− 4

5 − 3 5i

  • − 4

5 − 3 5i

−1 √−1

  • K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

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SLIDE 19

3.4: Inverse trigonometric/hyperbolic functions 3.5: Exponential and power functions 3.6: Branch points, branch cuts Branch points Branch cuts An example

√z, continue the walk

After going round the circle once more we end up at

  • ur 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

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SLIDE 20

3.4: Inverse trigonometric/hyperbolic functions 3.5: Exponential and power functions 3.6: Branch points, branch cuts 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

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SLIDE 21

3.4: Inverse trigonometric/hyperbolic functions 3.5: Exponential and power functions 3.6: Branch points, branch cuts 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 2π (positive or negative) hence log z has changed by 2πi. 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

  • ne takes the negative real axis.
  • K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

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SLIDE 22

3.4: Inverse trigonometric/hyperbolic functions 3.5: Exponential and power functions 3.6: Branch points, branch cuts Branch points Branch cuts An example

The arctangent function

arctan z is a composition z → 1 + iz 1 − iz → 1 2i log 1 + iz 1 − iz

  • The bilinear map maps i to 0 and −i to ∞, which are branch

points of log z. So i and −i are branch points of arctan z. The imaginary axis is mapped onto the real line. The triple (i, ∞, −i) is mapped to (0, −1, ∞).

  • K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

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SLIDE 23

3.4: Inverse trigonometric/hyperbolic functions 3.5: Exponential and power functions 3.6: Branch points, branch cuts Branch points Branch cuts An example

The arctangent function

The branch points and a branch cut for arctan z z i −i ∞

1+iz 1−iz

∞ −1

  • K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

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SLIDE 24

3.4: Inverse trigonometric/hyperbolic functions 3.5: Exponential and power functions 3.6: Branch points, branch cuts Branch points Branch cuts An example

The arctangent function

The branch points and a few more branch cuts for arctan z z i −i −1 1

1+iz 1−iz

∞ −i i

  • K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

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SLIDE 25

3.4: Inverse trigonometric/hyperbolic functions 3.5: Exponential and power functions 3.6: Branch points, branch cuts Branch points Branch cuts An example

What to do?

From the book: 3.4, 3.5, 3.6. Suitable exercises: 3.15 - 3.37 Recommended exercises: 3.19, 3.21, 3.22, 3.23, 3.27, 3.36, 3.37.

  • K. P. Hart

wi4243AP/wi4244AP: Complex Analysis